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Full text of "CANDE-1980 : box culverts and soil models"

FE 
562 

. K3 

no. 

r iWA- 

RD- 
80-172 



Report No. FHWA/RD-80/172 



IANDE-1980: BOX CULVERTS AND SOIL MODELS 



May 1981 
Final Report 




*Mm o* 



Document is available to the public through 
the National Technical Information Service, 
Springfield, Virginia 22161 



Prepared for 

FEDERAL HIGHWAY ADMINISTRATION 
Offices of Research & Development 
Structures and Applied Mechanics Division 
Washington, D.C. 20590 



FOREWORD 

This report presents the results of a study designed to extend the 
capability of the FHWA "CANDE" (Culvert Analysis and Design) computer 
program to include the capability for the automated finite element 
analysis for the structural design of precast reinforced concrete box 
culvert installations. The study also resulted in a new reinforced 
concrete model with loading through ultimate, unloading and redistribution 
of stresses due to cracking, as well as a new soil model (the so-called 
Duncan model). Included in the report is a User Manual Supplement and 
three (3) solved sample problems. Overlay instructions permit the 
program to be executed more efficiently and with less computer core 
storage requirements. This report will be of primary interest to 
supervisors, engineers, and consultants responsible for the design of 
culverts. 

This report is being distributed under FHWA Bulletin with sufficient 
copies of the report to provide one copy to each regional office, one 
copy to each division, and one copy to each State highway department. 
Direct distribution is being made to the division offices. 




"Charles F. Scheffey 
"j x Director, Office of Research 

/ Federal Highway Administration 



NOTICE 

This document is disseminated under the sponsorship of the Department of 
Transportation in the interest of information exchange. The United 
States Government assumes no liability for its contents or use thereof. 

The contents of this report reflect the views of the authors, who are 
responsible for the facts and the accuracy of the data presented herein. 
The contents do not necessarily reflect the official views or policy of 
the Department of Transportation. 

This report does not constitute a standard, specification, or regulation, 

The United States Government does not endorse products or manufacturers. 
Trademarks or manufacturers' names appear herein only because they are 
considered essential to the object of this document. 






Technical Report Documentation Page 






1. Report No. 

FHWA/RD-80/172 



2. Government Accession No. 



3. Recipient's Catalog No. 



4. Title and Subtitle 

CANDE-1980: Box Culverts and Soil Models 



5. Report Date 

May 1981 



6. Performing Organization Code 



7. Author's) 

Katona, M.G, Vittes, P.P., Lee, C.H., and Ho, H.T. 



8. Performing Organization Report No. 



9. Performing Organization Name and Address 

University of Notre Dame 
Notre Dame, Indiana 46556 



10. Work Unit No. (TRAIS) 

3513-241 



11. Contract or Gront No. 

D0T-FH-1 1-9408 



12. Sponsoring Agency Name and Address 

Offices of Research and Development 
Federal Highway Administration 
U.S. Department of Transportation 
Washington, D.C. 20590 



13. Type of Report and Period Covered 

Final Report 

November 1978-October 1980 



DEPARTMENT 

TRANSPORTATION 



\$}Y Spc isoring Agency Code 



15. Supplementary Notes 



George W. Ring, Contract Manager, HRS-14 



JAN11198Z 



\— 



LiDHAnV 

16. Abstract 

The CANDE computer program, introduced in 1 970 Fur lilt! iJTTUctijral design and analysis 
of buried culverts, is extended and enhanced in this work effort to include options 
for automated finite element analysis of precast, reinforced concrete box culverts, 
and new nonlinear soil models. User input instructions for the new options, now 
operative in the CANDE-1980 program, are provided in the appendix of this report 
along with example input/output data. 

Comparisons between CANDE-1980 predictions and the elastic analysis/design method 
used to develop the ASTM C789 design tables for precast box culverts revealed the 
importance of soil -structure interaction which is not taken into account in the latter 
method. As a general conclusion, the ASTM C 789 design tables provide safe designs 
(conservative) providing that good quality soil is used for backfill. 

The so-called Duncan soil model, employing hyperbolic functions for Young's modulus 
and bulk modulus, is a new soil model option in CANDE-1980. Standard soil model 
parameters, established from a large data base of triaxial tests, are stored in the 
program and can be used by simply identifying the type of soil and degree of compac- 
tion. In a similar manner, simplified data input options have also been developed 
for the overburden dependent soil model. 

In addition to user input instructions and example input/output data, the appendices 
also provide overlay instructions to reduce computer core storage requirements. 



17. Key Words 

Culverts, Box Culverts, Soil Models, 
Soil Structure Interaction 



18. Distribution Statement 



No restrictions. This document is available 
through the National Technical Information 
Service, Springfield, Virginia 22161. 



19. Security Classif. (of this report) 



Unclassified 



20. Security Classif. (of this page) 



Unclassified 



21. No. of Pages 
214 



22. Price 



Form DOT F 1700.7 (8-72) 



Reproduction of completed page authorized 



ACKNOWLEDGMENTS 

Representatives of industry, state highway departments, universities 
and research groups have been very helpful in providing information and 
constructive comments for this research effort. A special thank you is 
extended to Dr. James M. Duncan of the University of California for 
providing data and details of his soil model, and to Dr. Frank J. Heger 
of Simpson Gumpertz and Heger Inc. who, along with representatives of 
the American Concrete Pipe Association, supplied experimental data for 
out-of-ground tests. Mr. Robert Thacker provided consultation on over- 
laying the CANDE program on the IBM computers and programming the metric 
version of CANDE-1980. 



TABLE OF CONTENTS 

p age 

CHAPTER 1 - INTRODUCTION 1 

1.1 Background 1 

1.2 Objectives 2 

1.3 Scope and Approach 2 

CHAPTER 2 - REVIEW OF PRECAST BOX CULVERTS 4 

2.1 Background 4 

2.2 Development and rational of ASTM precast 

box culvert standards 5 

CHAPTER 3 - REINFORCED CONCRETE MODEL 10 

3.1 Objective 10 

3.2 Assumptions, limitations and approach 10 

3.3 Basic formulation for beam-rod element 12 

3.4 Finite element interpolations 16 

3.5 Stress-strain relationships 19 

3.6 Section properties 23 

3.7 Incremental solution strategy 24 

3.8 Measures of reinforced concrete performance 26 

3.9 Standard parameters for concrete and reinforcement .... 29 

CHAPTER 4 - EVALUATION OF REINFORCED CONCRETE MODEL FOR 

CIRCULAR PIPE LOADED IN THREE-EDGE BEARING 31 

4.1 Preliminary investigations 31 

4.2 Experimental tests 33 

4.3 Analytical model and comparison of results 37 

CHAPTER 5 - EVALUATION OF REINFORCED CONCRETE MODEL FOR 

BOX CULVERTS LOADED IN FOUR-EDGE BEARING 49 

5.1 Experimental tests 49 

5.2 CANDE model 53 

5.3 Comparison of models with experiments 53 

53 

CHAPTER 6 - DEVELOPMENT OF LEVEL 2 BOX MESH 62 

6.1 Parameters to define the models 62 

6.2 Assumptions and limitations 67 

CHAPTER 7 - EVALUATION OF CANDE BOX-SOIL SYSTEM 71 

7.1 Sensitivity of soil parameters 71 

7.2 Comparison with test data - 79 



in 



TABLE OF CONTENTS (Continued) 



Page 

CHAPTER 8 - EVALUATION OF ASTM C789 DESIGN TABLES WITH CANDE 85 

8.1 Box section studies for dead load 85 

8.2 Box section studies with live loads Ill 

CHAPTER 9 - SOIL MODELS 115 

9.1 Duncan model representation of elastic parameters 117 

9.2 CANDE solution strategy for Duncan model 121 

9.3 Standard hyperbolic parameters 125 

CHAPTER 10 - SUMMARY AND CONCLUSIONS 132 



APPENDICES 

A - Details of reinforced concrete model 134 

B - CANDE-1980; User Manual Supplement 147 

C - Sample of input data and output 177 

D - System overlay 202 

REFERENCES 208 



IV 



CHAPTER 1 



INTRODUCTION 



1.1 BACKGROUND 

The CANDE computer program (Culvert ANalysis and Design) was first 
introduced in 1976 for the structural analysis and design of buried cul- 
verts (1,2,3). CANDE employes soil-structure interaction analysis and 
has a variety of options, such as; choice of culvert type (corrugated 
steel, corrugated aluminum, reinforced concrete, and plastic) and choice 
of analysis /design method (elasticity solution - level 1, automated finite 
element solution - level 2, and standard finite element solution - level 3) 
Other features include; linear and nonlinear culvert and soil models, in- 
cremental construction and soil-structure interface elements. 

Since its introduction in 1976, the program has been widely distri- 
buted and used by state highway departments, federal agencies, consulting 
firms, industry, research laboratories, and universities in the United 
States and Canada. User responses have been very favorable along with 
encouragement and suggestions for extending the program's capabilities. 
In particular, it is observed that reinforced concrete box culverts have 
dramatically increased in use during recent years. To analyze these with 
CANDE (1976 version) requires level 3 analysis with time consuming finite 
element data preparation. Prior to this work, the automated finite element 
level 2 analysis was restricted to round or elliptical pipes. Thus, a 
desirable program extension is a level 2 analysis for box culverts with 
the capability to analyze through ultimate loading. A second observation 
is the wide spread popularity of the so-called Duncan soil model (26, 27, 
28, 29) which has been developing over the last decade and is formulated 
on a large experimental data base for many types of soil. The above 
observations lead to the objectives of this work. 



1.2 OBJECTIVES 

The first major objective is to develop and incorporate into the 
CANDE program an automated finite element analysis solution method for 
buried, precast reinforced concrete box culverts, called here, "level 2 
box" option. Included in this objective is validating the CANDE model 
with experimental data for loadings through ultimate and comparisons 
with other design/analysis methods. 

The second major objective is to incorporate the Duncan soil model 
into the CANDE program with due regard to convergence problems and to 
provide options for simplified data input. 

1.3 SCOPE AND APPROACH 

To meet the above objectives, a step by step approach was undertaken 
for both major goals. First, for the development and validation of pre- 
cast reinforced concrete box culverts, the steps are: 

(a) Review current design/analysis procedures to assess the state- 
of-the-art and to establish a comparitive basis with CANDE 
(Chapter 2). 

(b) Reformulate the existing reinforced concrete model to include 
loading through ultimate, unloading, and redistribution of 
stresses due to concrete cracking (Chapter 3 and Appendix A). 

(c) Evaluate and validate the reinforced concrete model with out- 
of-ground experimental data including pipes with 3-edge bearing 
loads and boxes with 4-edge bearing loads (Chapters 4 and 5). 

(d) Develop an automated finite element solution method (level 2 
box) for buried box culverts (box-soil model) with simplified 
input for embankment and trench installations (Chapter 6). 

(e) Evaluate and validate the box-soil model with available experi- 
mental data and parametric studies (Chapter 7). 

(f) Cross check the box-soil model predictions with current design/ 
analysis procedures in step (a) and evaluate current design 
methods (Chapter 8). 



Next, for the objective of incorporating the Duncan soil model and 
simplifying soil model input, the steps are (Chapter 9): 

(a) Evaluate the Duncan soil model to verify reasonable behavior 
in confined compression and triaxial loading. 

(b) Investigate iterative solution strategies to enhance convergence 
and incorporate the model into CANDE program. 

(c) Establish standard model parameters dependent on soil type and 
degree of compaction for the simplified data input option. 
Also, simplify data input for the existing overburden dependent 
soil model. 

All program modifications noted above have been incorporated into 
CANDE, hereafter called CANDE-1980 to distinguish it from the 1976 version. 
Appendix B provides input instructions to exercise the new options contained 
in CANDE-1980. These instructions are a supplement to the 1976 CANDE User 
Manual (2) and only need to be referred to if the new options are desired. 
In other words, the 1976 user manual is compatible with the CANDE-1980 
program. Appendix C illustrates input-output data for some of the new 
options and Appendix D provides system overlay instructions to reduce 
core storage. 

The CANDE-1980 program discussed herein is based on the English 
system of units. A companion program in metric units has been developed 
and is also available from FHWA. 



CHAPTER 2 
REVIEW OF PRECAST BOX CULVERTS 

In this chapter a brief review on the development of precast rein- 
forced concrete box culverts is presented along with a discussion of 
current design procedures. The intent is to acquaint the reader with 
precast box culverts, terminology and design concepts and to "set the 
stage" for the CANDE methodology presented in later chapters. For 
brevity, "reinforced concrete box culverts" will be referred to as "box 
culverts". 

2.1 BACKGROUND 

Precast box culverts, as opposed to cast-in-place box culverts, are 
relatively recent additions in culvert technology, coming into popular 
use within the last decade. For many years, cast-in-place box culverts 
have been used in installations with special requirements or by design 
preference. However, cast-in-place culverts have inherent disadvantages; 
high labor costs associated with cast-in-place construction, lengthy 
periods of traffic disruption, and minimal quality control often compensated 
for by conservative designs. Alternatively, plant-produced box culverts, 
manufactured under strict quality control and installed by rapid cut-and- 
fill procedures, can offset these disadvantages particularly if the box 
dimensions, reinforcement, ect., are standardized for manufacture. 

With the above motivation, the Virginia Department of Highways 
and the American Concrete Pipe Association (ACPA) , with financial support 
from the Wire Reinforcement Institute, initiated a cooperative program, 
early in 1971, to develop manufacturing specifications and standard 
designs for precast box culverts. These specifications were to be 
adaptable as a national standard under the auspices of the American 
Society of Testing Materials (ASTM) and the American Association of State 
Highway and Transportation Officials (AASHTO) . To this end, ACPA contracted 



the consulting firm of Simpson, Gumpertz and Heger Inc. (SGH) to develop 
a computerized design program for precast box culverts in cooperation 
with ASTM committee C-13. 

Ultimately, this effort culminated in the ASTM C789 and AASHTO M259 
specifications on Precast Reinforced Concrete Box Sections for Culverts, 
Storm Drains and Sewers first published in 1974. These specifications were 
limited to box culverts with a minimum of two feet (0.61 m) of earth cover. 
Further developmental work by SGH resulted in the additional specifications 
ASTM C850 and AASHTO M273 published in 1976 for precast box culvert in- 
stallations with less than two feet of earth cover. The above ASTM and 
AASHTO specifications are essentially the same except for a few details 
which are apparently now resolved. For purposes of this study, the 
ASTM specifications will be used as reference. Design methods for pre- 
cast box culverts, other than those embodied in ASTM or AASHTO specifi- 
cations, will not be reviewed here since they are not standardized nor 
have they gained national acceptance. Recently, ACPA published a sur- 
vey (Concrete Pipe News, June 1980) showing that usage of precast box 
culverts, designed by ASTM specifications, has increased dramatically 
within the last year. . . the number of projects and linear footage 
installed in 1979 is almost equal to the total for the previous five years! 

2.2 DEVELOPMENT AND RATIONALE OF ASTM PRECAST BOX CULVERT STANDARDS 

The SGH computerized design program (12) is the basis of the 
design rationale in the ASTM C789 design tables. A typical box cross- 
section is shown in Figure 2.1 along with nomenclature. The SGH design/ 
analysis approach includes the following steps; (a) load distributions 
are assumed around the culvert in an attempt to simulate dead earth loads 
and live loads, (b) moment, shear, and thrust distributions are determined 
by standard matrix methods using elastic, uncracked concrete section pro- 
perties, (c) in the design mode, steel areas are determined by an ulti- 
mate strength theory for bending and thrust, where ultimate moments and 
thrusts are obtained from step (b) multiplied by a load factor, (d) crack- 
width (0.01 inch allowable) is checked using a semi-empirical formula 



M 



^1 



f 


^ 


1 


/\ 


T 1 

4> 


-v At2 \ 


-» 


<-M 


R 

:-A s4 


1 


_, 



k s1 



<-C 



t. 

T 



A s i = outer reinforcement 

A S 2= top inner reinforcement 

A S 3= bottom inner reinforcement 

A S 4= side inner reinforcement 

C = cover distance of reinforcement, uniform 

H=haunch dimension 

M=minimum length A s1 steei, top and bottom 

R=rise distance , inside to inside 

S = span distance, inside to inside 

T=wall thickness, uniform 



Figure 2.1. Typical Box Culvert Cross Section 



controlled by steel stress at service loads, (e) ultimate shear stress 
( 2 /P~ ) is checked against the nominal shear stress obtained in step (b) 
multipled by a load factor. 

For the standard box sizes shown in Table 2.1, the SGH design program 
was used to generate the ASTM C789 design tables wherein steel reinforce- 
ment requirements are specified as a function of design earth cover be- 
ginning with a two foot minimum. 

In a similar manner, ASTM C850 design tables were generated for earth 
covers less than two feet. Here, the SGH design procedure was modified 
to include requirements for longitudinal steel design due to concentrated 
live loads (see ASTM Symposium STP 630). 

Although the SGH design/analysis program has not been validated 
with experimental data from buried box culverts, fairly good correlation 
with out-of-ground experimental tests has been reported (13). More will 
be said about these experiments in Chapter 5. 

Experimental data for instrumented, buried box culverts is extremely 
limited. As of this writing, only two state highway departments (Kentucky 
and Illinois) are known to have undertaken experimental programs for in- 
strumenting (settlement, soil pressure, and strain gages) buried box 
culvert installations. Other states have made visual inspection reports 
on the performance of buried box installations, but this data has marginal 
value for validating design/analysis procedures. Data from the Kentucky 
Department of Transportation was made available for this study and is 
used to evaluate the CANDE program in Chapter 7. 

In summary, the ASTM design tables for buried, precast box culverts, 
which are based on the SGH design/analysis program, have not been pre- 
viously validated with experimental data from buried installations. Nor 
have the tables been cross-checked with analytical procedures, such as 
CANDE, employing soil-structure interaction and the nonlinear nature of 
reinforced concrete. With this goal in mind, a step by step approach is 
presented in the following chapters. First, the theory of CANDE 's non- 
linear, reinforced concrete model is developed. Second, the model is 



TABLE 2.1 Standard box sizes, ASTM C789 



Span 
ft. 


2 


3 


4 


5 


Rise, ft. 
6 7 8 


9 


10 


Wall 
Thickness 
in. 


3 


X 


X 












4 


4 


X 


X 


X 










5 


5 




X 


X 


X 








6 


6 




X 


X 


X 


X 






7 


7 






X 


X 


X X 






8 


8 






X 


X 


XXX 






8 


9 








X 


XXX 


X 




9 


10 








X 


XXX 


X 


X 


10 



1 ft = 0.3048 m 
1 in = 2.54 cm 



validated with experimental data for out-of-ground conditions. Third, 
the reinforced concrete model is combined with soil system models and 
compared with experimental data from a buried installation. Last, the 
CANDE model is used to evaluate the ASTM design tables. 



CHAPTER 3 
REINFORCED CONCRETE MODEL 

3.1 OBJECTIVE 

A reinforced concrete, beam-rod member, whether it be part of a 
culvert or any other structural system, poses a difficult analysis 
problem due to the nonlinear material behavior of concrete in com- 
pression, cracking of concrete in tension, yielding of reinforcement 
steel, and the composite interaction of concrete and reinforcement. 
Matters are further complicated when the internal loading is not 
proportional, i.e., when the internal moment, shear and thrust at 
a particular cross section change in different proportions (including 
load reversals) during the loading history. Such is the case for 
buried culverts during the installation process. 

In this chapter, the development of a reinforced concrete beam- 
rod element is developed in the context of a finite element formulation 
for CANDE-1980. This model is more general than the model in CANDE-1976 
and includes; incremental loading through ultimate, unloading, and 
redistribution of stresses due to cracking. 

The following presentation provides an overview of the model 
development emphasizing assumptions and limitations. Details of the 
numerical solution strategy are presented in Appendix A. Evaluation 
of the model with experimental data and other theories is presented 
in subsequent chapters. 

3.2 ASSUMPTIONS, LIMITATIONS, AND APPROACH 

Listed below are the fundamental assumptions for the reinforced 
concrete beam-rod element. 



10 



I 



1. Geometry and loading conform to plane strain implying 
the beam- rod element is of unit width. Constant section 
properties are assumed through an element length, but 
may differ between elements. 

2. Displacements and strains are small. No buckling consid- 
erations are included. 

3. Planes remain plane in bending and shear deformation is 
negligible. 

4. Concrete is linear in tension up to cracking. Cracked 
concrete cannot carry tension stresses and pre-crack 
stresses are redistributed. In compression, concrete is 
modeled with a trilinear stress-strain curve terminating 
at ultimate strain. Unloading is elastic. 

5. Reinforcement steel is elastic-plastic and identical in 
compression and tension. Unloading is elastic. 

6. Reinforcement steel is lumped into two discrete points 
near the top and bottom of the cross-section and deforms 
with the cross-section, 

7. Element lengths are sufficiently small so that the current 
stress distribution through a cross-section is representative 
of the entire element for purposes of computing current 
section properties. 

8. Loads are applied incrementally and sufficiently small 
so that the stress-strain relations (for both steel and 
concrete) can be regarded as incremental tangent relations 
determined iteratively over the load step. 



11 



In overall perspective, the developmental steps begin with an 
incremental statement of virtual work wherein the beam-rod assumptions 
are introduced along with standard finite element interpolation func- 
tions for axial and bending deflections. This results in a tangent 
element stiffness matrix and incremental load vector that can be 
assembled into a global set of system equations with unknown nodal 
degrees of freedom, and solved by standard techniques (1). However, 
the global matrix contains estimates of the bending and axial stiff- 
ness for each beam-rod element (as well as estimates for soil stiff- 
ness if nonlinear soil models are part of the system). Thus, each 
load step is repetitively solved (iterated), and the results are used 
to improve the stiffness estimates until convergence is achieved. 

Prior to the first loading increment, the beam-rod element is 
assumed stress free and uncracked so initial stiffnesses correspond 
to an uncracked, elastic, transformed reinforced concrete cross- 
section. Upon applying the first load increment, the first tentative 
solution may indicate that some elements should have had reduced 
stiffnesses due to cracking or yielding of the section. Using the 
strain distribution at the beginning and end of the load step, new 
stiffness estimates are obtained and the process is iterated to con- 
vergence. Each subsequent load step is treated in a similar fashion 
where a history of maximum stress and strain is maintained for pur- 
poses of identifying unloading conditions. 

The above assumptions and general approach are outlined in the 
following development, 

3,3 BASIC FORMULATION FOR BEAM- ROD ELEMENT 

In this section we consider an incremental virtual work statement 
for a unit width, beam-rod element with body forces given by: 

6AV = 5AU - 6AW (3,1) 

w ith 5AU = / / SeAa dxdy = internal virtual work increment 

x y 



12 



SAW = f f 6{. 7 } { , -1} dxdy = external virtual work increment 
x y v Af„ 

where a = normal stress, x-direction 

e = normal strain, x-direction 

u = longitudinal displacement, x-direction 

v = transverse displacement, y-direction 

f. = longitudinal body force, x-direction 

f_ = transverse body force, y-direction 

x = space coordinate parallel to beam axis 

y = space coordinate transverse to beam axis 

6 = virtual symbol 

A = increment symbol 

The above beam displacements are illustrated in Figure 3.1. 

Introducing Bernouli-Euler beam kinematics (Assumptions 2 and 3), 
longitudinal displacements through a cross section may be arbitrarily 
decomposed into a uniform axial distribution, u (x), plus a distri- 
bution proportional to slope, v'(x), and linearly varying about some 
axis y, i.e. : 

u(x,y) = u Q (x) + v f (x) (y-y) 3.2 

Later, when the above kinematic relation is incorporated into Equation 
3,1, the axis y will be chosen such that internal bending work is un- 
coupled from internal axial work, 

Employing the small strain-displacement assumption, normal strain 
is: 

e(x,y) = u^(x) + v"(x) (5-y) 3.3 

where primes denote derivatives with respect to the argument. 

To complete the field variable assumptions, a general, nonlinear 
stress-strain relationship is assumed in incremental form as: 

Aa = E'(e)Ae 3,4 



13 



4* 




Figure 3.1 Deformation of Beam-Rod Element 






AF- 



A t 1 MHt t f t f t t t 

Ui /^R~ AF1 



A 



i>r«i 






£>-* 



e 



i 



Figure 3.2 Nodal Degrees of Freedom 
and Element Loading 



14 



Here E f (e) is a tangent modulus relating increments of stress to 
increments of strain and is dependent on loading history. Naturally, 
the functional forms of E'(e) are different for concrete and steel 
materials. However for clarity of presentation, the specific forms of 
E'(e) will be deferred to a later section. 

Using the incremental form of Equation 3.3 along with Equation 3.4 
and integrating through the cross section, the internal virtual work 
increment may be expressed as: 

5AU = / (6ui EA* Aul + 6v M EI* v" + EX*(5v"Au: + 6ulAv"))dx 3.5 
x 

* 

where EA = / E' (e) dy = effective axial stiffness 3.6 

y 

* _ 2 

EI = / E'(e) (y-y) dy = effective bending 

y stiffness 3.7 

* 

EX = / E'(e) (y-y) dy = axial-bending coupling 3.8 

_ * 

The location of y is now chosen so that the coupling term EX is 

zero. Thus, y is given by: 

y = ( / E»(e) y dy)/EA* 3.9 

This choice of y is convenient because bending and axial deformations 
are uncoupled in the virtual work statement. However, it must be 
remembered that y, like EA and EI , is dependent on E'(e), thus these 
values change during each load step. 

To complete the virtual work statement, the kinematic assumption 
(Equation 3.2) is introduced into the external virtual work incremen- 
tal expression and integrated over the cross section to give: 

6AW = / ( 6u AF l + 6vAF 2 + <Sv?AF 3 ) dx 3 ' 10 

where AF = / Af, dy = axial body force per unit length 

1 y 1 

AF„ = J" Af dy = transverse bodv force per unit length 

2 y 2 J f 



15 



AF = / Af (y-y) dy = body moment per unit length 
■j y x 

The body moment, AF , is generally nonzero except if the centroid of 
the axial body weight happens to coincide with the current location 
of y. However, the magnitude of the body moment is usually negligible 
compared to the magnitude of internal moments which arise from trans- 
verse loading in culvert installations. Thus, the body moment is 
neglected in this study. 

Equations 3.5 and 3.10 are the internal and external virtual work 
expressions for the beam- rod element with unknown displacement functions 
u (x) and v(x). 

3.4 FINITE ELEMENT INTERPOLATIONS 

Figure 3.2 shows a beam-rod element with three nodal degrees of 
freedom at each end node, an axial displacement, a vertical displace- 
ment, and a rotation. These degrees of freedom are used to define 
admissible interpolation functions for u n (x) and v(x) in the context 
of a finite element formulation. 

Specifically, the axially displacement, u (x) is approximated 
with a two-point Lagrange interpolation function: 



■■ * D] 



u Q (x) = O, cj) > 3.11 



where u, = axial displacement at node 1 



u~ = axial displacement at node 2 
(J> 1 (x) =1-6 
4> 2 (x) = 3 

B(X) = x/£ 



For transverse displacements, v(x), a two-point Hermetian inter- 
polation function is used. 



16 



v(x) = 



<a l a 2 a 3 a ? 



v. 



3.12 



where 



v = transverse displacement at node 1 



v„ = transverse displacement at node 2 



6.. = rotation at node 1 



6„ = rotation at node 2 



a 1 (x) 



2 3 

1 - 33 + 23 



a 2 (x) 



a 3 (x) 



= 3(1-3) I 



2 3 

33 - 23 



a 4 (x) 



= 3 (3-D I 



Upon substituting the interpolation functions into the Incremental 



virtual work expression, 6AV = 5AU-6AW, we have: 

e 



6AV = <6r > {[K ] {Ar} - {AP }} 
e e e 



3.13 



where 



{r} 



~* A 


*■■ 


u l 




* 




v 1 

J_ 




A 




6 1 




/v 




U 2 




* 




9 2 




/v 




_ v 2 





= element degrees of freedom 3.14 



17 



{AP } = -jV 
e 12 



6AF ] 
6AF, 

JIAF, 

i 

6AF ] 
6AF, 

-£AF, 



= element load vector 



3.15 



[K ] = 

e 



* * 

EA n n -EA 



\J 


£ 


■k 

12EI 


6EI 


* 3 


* 2 




* 
«. 


c. 


* 

EA 



Symmetric. 



(tangent element 
stiffness matrix) 



-12EI 



-6EI 



6EI 



2EI 



* * 

12EI -6EI 



4EI 



The above tangent element stiffness and load vector are valid for 
the local beam coordinates. For assembling element contributions into 
the global coordinate system, standard coordinate transformation are 
employed. 

Note that the tangent element stiffness matrix is identical in 
form to that obtained from standard matrix methods of structural anal- 
ysis. However, the axial stiffness EA and bending stiffness EI 
(dependent on y) are not constant and must be determined iteratively 
for each load step in accordance with Equations 3.6, 3.7, and 3.9. 
These equations are dependent on the concrete and steel stress-strain 
relationships discussed next. 



18 



3.5 STRESS-STRAIN RELATIONSHIPS 

Concrete . The assumed stress-strain behavior for concrete is shown 
in Figure 3.3 where the trilinear curve is defined by the following 
input variables: 

e = concrete strain at initial tensile cracking 

e = concrete strain at initial elastic limit 

y 

e' = concrete strain at onset of ultimate 
c 

f ' = unconfined compressive strength of concrete 
E = Young's modulus in linear zone 

With the above input variables, three additional parameters can be 
derived: 

E n = (f - E e )/(e' - e ) = Young's modulus in yielding 
2 c 1 y c y 

J zone 

£' - E, e = initial tensile strength 
tit 

f = E,e = initial yield strength 

yc 1 y J b 

In tension the concrete is linear until the initial tensile strain 
exceeds the cracking strain limit e . When cracking occurs, the 
tensile stress becomes abruptly zero (redistributed to noncracked 
portions). Once a point in the cross section is cracked, the crack 
does not heal, implying no tensile strength. Thus e is set to 
zero for all subsequent reloading in tension. 

For initial compression loading, the concrete begins to yield 
with hardening at stress f . Perfect plasticity occurs at stress f^ 
and continues through ultimate strain. Unloading is elastic and 
results in permanent plastic strains as indicated in Figure 3.4. 
Reloading is elastic until the stress reaches its previous maximum 
value after which it follows the original stress-strain curve. (See 
Figure 3.4). 



19 



CO 
(0 




Cracking 



> Strain 



Figure 3.3 Idealized Stress-Strain Diagram for Concrete 





unload-reload paths 



i/- iE i A 

/ / 

/ 

/ 



-^Strain 



Figure 3.4 Elastic Unload-Reload for Concrete 



20 



With the above understanding, the tangent modulus relationship 
for concrete confined in a plane is expressed as: 

E'(e) = E (1 - a(e)) 3.17 

c c 

2 
where E = E / (1 - v ) 
c 1 c 

with E = elastic, confined plane modulus of concrete 

v ■ Poisson's ratio of concrete (constant) 
c 

a(e) = dimensionless function of stress-strain history 



The dimensionless function a(e) ranges in value from 0.0 (elastic 
response) to 1.0 (perfectly-plastic response), representing the non- 
linear effect of concrete. The actual value of a(e) to be used for 
any given load increment is dependent on; known values of stress and 
strain at the beginning of the step, known history parameters for 
cracking and yielding, and unknown values of stress and strain at the 
end of the step (iteration). Appendix A provides the details for 
determining a(e) for all loading histories. 

Steel . The assumed stress-strain behavior for reinforcing steel 

is shown in Figure 3.5 where the elastic-plastic curve is characterized 

with two input variables: 

E n = Young's modulus for steel 

f = steel yield strength 

Behavior in compression and tension is identical so that material is 
elastic whenever the stress magnitude is less than f . Nonhardening 
plastic flow occurs when the stress is equal to f , Unloading from 
the plastic range is elastic and results in permanent plastic strains 
(see Figure 3,5). 

Similar to Equation 3,17 for concrete, the tangent modulus relation- 
ship for reinforcement steel confined in a plane is expressed as: 



21 



(0 

© 



CO 



fy-~ 




s 



s 



-fy 



-^Strain 



Figure 3.5 Idealized Stress-Strain Diagram 
for Reinforcing Steel 




X 



unit I 
width 7 ' 



Figure 3.6 Reinforced Concrete Cross Section 



22 



E' (e) = E (1 - a(e)) 3.18 

s s 

2 
where E = E J (1 - v ) 
s s 

with E = elastic, confined plane modulus of steel 
s 

v = Poisson's ratio of steel (constant) 
s 

a(e) ■ dimensionless function of stress-strain history 

As in the case of concrete, the function a(e) for steel ranges in 
value from 0,0 (elastic) to 1.0 (perfectly plastic) depending on 
stress-strain history and stress values at the beginning and end of 
each load step (see Appendix A). 



3.6 SECTION PROPERTIES 

Equations 3,17 and 3.18 represent the tangent modulus relation- 
ships for concrete and steel, respectively, which now can be used 
to evaluate current section properties EA , y, and EI , 

Referring to a typical cross section shown in Figure 3.6, the 
effective axial stiffness (Equation 3.6), the bending axis (Equation 
3.9), and the effective bending stiffness (Equation 3.7) can be ev- 
aluated by separating the concrete and steel integration areas as 
shown below. 



* A 



EA = 



E'(e) dy + A . E» (e . ) + A _ E» (e A ) 3.19 

c si s i sO s 



y = <[ E»( £ ) ydy + A . E' (e.) y. + A . E' (e )y )/EA* 3.20 
J j c v ' J si s i J x sO s 



EI = 



rh 2 _.,.,- ,2 







E* (?) (y-y) dy + A . E' (e . ) (y-y.) 
c si si i 



+ a so e ; < i o> (? - y o )2 3 - 21 



23 



where A . = bottom steel area per unit width 
. si 

A - = top steel area per unit width 
sO 

y. = distance to A . from bottom 
y i sx 

y^ = distance to A _ from bottom 
J sO 



The integrals containing E'(e) represent the concrete contribution 

c 

to section properties and are evaluated numerically with 11-point 
Simpson integration, A stress-strain history is maintained at each 
integration point for determining the current values of a(e). Steel 

contributions to section properties are governed by E' (e ) and E'(e ) 

S X s u 

representing the tangent steel modulus at the centroid of bottom and 
top steel reinforcement. 

The above equations suggest that the concrete contributions are 
integrated over the entire section area irrespective of "holes" where 
steel exists, however, the algorithm used in this study accounts for 
these holes. These and other details of computing section properties 
are discussed in Appendix A, 

3.7 INCREMENTAL SOLUTION STRATEGY 

All the assumptions and derivations for the beam-rod element 
have been presented. An overview of the solution strategy is given 
next. 

It is assumed that a converged solution is known at load step 

i-1 and it is desired to obtain a converged solution at load step i. 

Basically, the objective is to determine effective section properties, 

* - * 
EI , y, and EI for each beam- rod element. 

A flow chart of the solution strategy is illustrated in Figure. 3. 7. 
The procedure begins by initially assuming the section properties are 
the same as the previous load step. Next, the system is assembled 
for the current load increment and trial solutions are obtained for 
moment and thrust increments in each element, given by: 



24 



* A 

Estimate EA , y, EI for each element from 
load step i-1. 



Apply load increment and solve system. 
Obtain trial moment, AM, and thrust, AN, 
increments for load step i. 



3. 




Estimate new strain distribution at load 
step i as: 



AN 

= e ■ -i + — * 

l-l * 

EA 



+ — * (y-y) 

EI 



No 



No 



5. 



* _ 



4. Compute new estimates for EA , y, and EI 
(Equations 3.19, 3.20, and 3.21). 



Test for inner loop convergence, i...e._, , 
Are two successive estimates of EA , y, and EI 
(computed in Step 4) equal? 



yes 



6. Compute moment and thrust that must be 
redistributed due to cracking. 



Test for outer loop convergence, i.e., Are two 
successive estimates of EA , y, and EI used in 
Step 2 equal? If not, return to Step 2 and in Step 3 
add the effects of redistribution (first time only). 



yes 



8. Converged solution increment. Sum incremental 
responses to total response. Advance the load 
step (i -> i+1) and return to Step 1. 



Figure 3.7. Flow chart of solution strategy. 



25 



AN = /. Aa dA = EA Au' 
A. 

AM - f Aa (y-y) dA - EI Av" 

A 

Using the above relations together with Equation 3.3, a new strain 
distribution is estimated as shown in Step 3 of the flow chart. This, 
in turn, permits improving the estimates for section properties in the 
"inner loop" iteration; steps 3, 4, and 5. Here, AM and AN remain 
fixed (as estimated in Step 2) while the corresponding section 
properties are determined. Note that inner loop operations are at 
the element level, requiring no global assembly or solution. 

Each time the inner loop converges, the converged section proper- 
ties are used in Step 2 to get new global solutions for AM and AN. 
This process is called "outer loop" iteration and continues until two 
successive solutions are equal within a specified tolerance. When 
this occurs, convergence is achieved and the program advances to the 
next load step (see Appendix A for additional detail), 

3.8 MEASURES OF REINFORCED CONCRETE PERFORMANCE 

Once a converged solution is obtained, measures of structural 
distress are assessed by; (a) maximum tensile stress in steel (b), 
maximum compressive stress in concrete, (c) maximum shear stress in 
concrete, and (d) maximum crackwidth in concrete. The first three 
measures of distress are evaluated directly from the structural 
response predictions from the CANDE model, however the crackwidth 
prediction employs a semi-empirical approach. Each distress measure 
is normalized by a corresponding design criterion to produce perform- 
ance factors as discussed below. 

Steel Tension . The performance factor for steel reinforcing is 

given by: 

PF , = f /f 
steel y max 

where f = maximum steel stress (predicted) 
max r 

f = steel yield stress 

y 



26 



For properly designed structures, this performance factor should be 
in the range of 1,5 to 2,0, When the steel begins to yield, the 
performance factor becomes 1,0 and remains .there through ultimate 
loading. 

Concrete Compression . For the outer concrete fibers experiencing 
compressive stress from thrust and bending, the performance factor is: 

PF = V/o 
comp . c max 

where a = maximum compressive stress (predicted) 

f* = compressive strength of concrete 
c 

Proper designs should have this performance factor in the range 1.6 to 
2.5, The performance factor remains at 1.0 when the concrete becomes 
perfectly plastic and remains there through ultimate loading. 

Concrete Shear . Nominal shear stress through a cross section is used 
to define the shear performance factor, given by: 

PF . o v /v 
shear c max 

where v = nominal average shear stress on section 
max ° 

v = nominal concrete shear strength 
c 

Here v is computed by dividing the maximum predicted shear force by 

the concrete area minus the cover area of steel. This definition is 

consistent with the standard ACI measure of shear strength for beams 

given by: 

v - 2.0 TV (psi) 
c c 

Other measures of shear strength are examined in the next chapter 

with experimental data, 

For proper design, the above performance factor should be in 
the range 1.7 to 2.7. In the absence of stirrups, shear failure 
(e.g. diagonal cracking) is assumed to occur when the performance 



27 



factor value is 1.0. Note that the CANDE model does not incorporate 
diagonal cracking into the stress-strain law, only flexural cracking. 

Concrete Crackwidth . The crackwidth prediction, C , is a semi- 
empirical approach wherein the maximum tensile steel stress predicted 
by CANDE is used in an empirical formula proposed by Gergely and Lutz 
(10). Using 0.01 inches (0.0254 cm) as the design standard for allowable 
crackwidth, the cracking, performance factor is defined as: 

PP , » 0.01/C 
crack w 



3 



/£? 



where C - Q.091/2t* S (f - 5000)R (inches) 

W D S 

R ■ 1,34 x 10 (dimensionless number for culvert slabs). 

t. ■ concrete cover to steel centroid (inches) 

D 

f ■ tensile steel stress (psi) 

S ■ spacing of reinforcement (inches) 



The Gergely and Lutz formula for C was found to give good pre- 

w 

dictions for crackwidths in this study. This finding is further 
supported by Lloyd, Relaji and Kesler (11) in their experimental tests 
on one-way slabs with deformed wire, deformed wire fabric, and deformed 
bars. The new crackwidth formula defined above replaces the old 

crackwidth formula in CANDE- 19 76. The new formula can be made to be 

2 
identical to the old by defining S = 0.68/A t^ where A is tension 

2 sos 

steel reinforcement, in /in. 



Ultimate Loads . Ultimate loading in thrust and bending occurs in 
a .beam- rod element when the reinforced concrete section cannot sustain 
any additional loading, i.e., all uncracked concrete is at maximum 
compressive strength f ' and all reinforcement steel is yielding (plastic 
hinging). For a structure composed of beam-rod elements, such as a 
box culvert, ultimate loading occurs when a sufficient number of plastic 
hinges have formed to produce a collapse mechanism. This can be de- 
termined from the CANDE program by observing unrestrained deformation 
as the load is increased to ultimate. 



28 



Ultimate loading in shear is assumed to occur when the performance 
factor for shear in any beam-rod element becomes 1,0. If a structure 
fails in shear prior to flexural-thrust failures, the CANDE model is 
still capable of carrying load up to flexural-thrust failure because 
diagonal cracking is not included in the model development. Thus for 
loads exceeding concrete shear failure, it must be presumed that suf- 
ficient shear reinforcement (stirrups) is available. 

3.9 STANDARD PARAMETERS FOR CONCRETE AND REINFORCEMENT 

Based on investigations presented in subsequent chapters, a set 
of standard parameter values for concrete is given in Table 3,1 (see 
also Figure 3.3). Except for compressive strength f and cracking 
strain e , the parameters are assigned unique values, some of which 
are dependent on f*.' 

For subsequent analytical studies, the concrete will be 

characterized by specifying f and £ . The remaining parameters 

c t 

are assigned the standard values shown in Table 3.1 unless stated 
otherwise. 

Standard parameters for reinforcement steel are shown in Table 3.2 
wherein the yield stress in considered as the primary variable. 



29 



Table 3.1 



Standard Concrete Parameters 



Parameter 



Symbol 



Value 



Compressive strength 

Elastic modulus 

Cracking Strain 

Initial yield strain 

Strain at f 
c 

Weight density 
Poisson's ratio 



f 
c 



3000 to 7000 (psi) 
33/F (y ) 1,5 (psi) 



c c 




0,0 to 0,0001 


(in/in) 


0.5 f /E, 
c 1 


(in/ in) 


0,002 


(in/in) 


150 


(lbs/ft 3 ) 


0.17 


_ 



Table 3.2 



Standard Steel Parameters 



Parameter 



Symbol 



Value 



Yield strengh 
Elastic modulus 
Poisson's ratio 



30 to 90 


ksi 


29000 


ksi 


0.3 


_ 



1 psi = 6.895 kPa 
1 pcf = 157.1 N/m~ 



30 



CHAPTER 4 

EVALUATION OF REINFORCED CONCRETE MODEL 
FOR CIRCULAR PIPE LOADED IN THREE-EDGE BEARING 

In this chapter the validity of the reinforced concrete model 
(presented in the previous chapter) is examined by comparing results 
with experimental data for circular pipe tested out-of-ground in three 
edge bearing, i.e., the so-called D-load test (ASTM C497-65T), The 
objective is to determine if the model can reasonably predict load- 
deflection histories, the load at which 0.01 inch (0.254 cm) crackwidths 
occur, and ultimate load. 

4.1 PRELIMINARY INVESTIGATIONS 

Prior to comparing the model performance with circular pipe test 
data, a preliminary study was undertaken for staticrlly determinate, 
reinforced concrete beams with transverse loading and combined trans- 
verse with axial loading. The purpose of this preliminary study was 
to investigate the sensitivity of modeling parameters and to compare 
the model predictions with published experimental beam data (8,9) and 
conventional ultimate strength theories (4,5). Major findings from 
the preliminary study are listed below, additional detail is reported 
in Reference (6) . 

1. For all the beams studied, including both single and double 
reinforcement, the predicted ultimate moment capacity for 
transverse loading agreed within 1% to those computed in 
accordance with ACI 318-77. 

2. Predicted load-deflection curves through ultimate were in 
close agreement with experimental data (8) obtained from 
two point loading of simply supported, rectangular beams 
with approximately 1.7% tension steel reinforcement. 



31 



3. In the presence of axial thrust loads, the predicted ultimate 
moment capacity was in good agreement with experimental data 
(9), wherein the ultimate moment capacity initially is in- 
creased as the axial thrust increased up to the balance 
point on the ultimate moment- thrust interaction diagram. 
Thereafter, the moment capacity steadily decreased to zero 

as thrust was increased to ultimate. 

4. As expected, the predictions for ultimate thrust-moment 

capacity were not influenced by the model input parameters 

e , e , and e' which describe the concrete stress-strain 
t y c 

curve up to compressive strength. Only the strength para- 
meters for concrete and steel (f ' and f ) influenced ultimate 

c y 

capacity. However, the load-deflection path to ultimate is 

influenced by e , e , and e 1 and the initial elastic moduli 
J t' y* c 

values for steel and concrete, 

5. The concrete cracking strain parameter e was found to have 
a significant effect on the load-deformation curves for 
lightly reinforced beams (typical for culvert cross-sections). 
As the parameter e decreases over a practical range (0.0001 
to 0.0) the effective stiffness decreases resulting in 
greater deformations for the same load. 

6. The compressive concrete strain parameters, e , and e', also 
influence the shape of the load-deformation curves, but to 

a lesser extent than e . As e is decreased over the range 

t y 

0.0008 to 0.0003 the deformations slightly increase. Con- 
versely, as e' is decreased over the range 0.0025 to 0.0015 
c 

deformations decrease. 



These preliminary studies demonstrated that the reinforced concrete 
model was working properly and provided insights for modeling and 
interpreting results for the circular pipes in three-edge bearing dis- 
cussed next. 



32 



4.2 EXPERIMENTAL TESTS 

The out-of-ground test results used in this study were obtained from 
an experimental study by Heger and Saba (15), wherein they tested rein- 
forced concrete circular pipes under three-edge bearing loadings as shown 
in Figure 4.1. Test results included; ultimate strength (load capacity), 
0.01 inch cracking l:>ad, deflections, visual observations of crack devel- 
opment, and stresses; in the reinforcing steel and in the concrete wall. 

The pipe test program consisted of 39 pipe specimens with different 
wall and diameter dimensions and amounts of reinforcement. For some 
pipes, stirrup reinforcement was used to prevent diagonal tension failure. 
The unconfined compressive strength of concrete was obtained using 
cylinder and core tests, the tensile strength of concrete was obtained 
with a split cylinder test, and the ultimate tensile strength, yield 
strength and modulus of elasticity for the steel wires were obtained 
with tests carried out in accordance with the ASTM Specification A185- 
56T for Welded Steel Wire Fabric. 

From the 39 pipes tested a subset of seven pipes are selected for 
this study. The subset represents the complete range of pipe dimensions 
and amounts of steel reiinforcement used in the test program. Table 4.1 
along with Figure 4.1 identifies the geometry of each selected pipe in 
three diameter groups; 48- inch, 72-inch and 108- inch pipes (1.22 m, 1.83 m 
and 2.74 m) . Each dianeter group has a constant wall thickness with 
different amounts of steel reinforcement. Ideally, each group should 
consist of low, medium, and high levels of steel reinforcement. However, 
the experiment did not include tests with medium levels of reinforcement 
for the 43- inch or 108- inch pipe. Thus all groups contain low and high 
reinforcement levels, but only the 72-inch pipe also has medium reinforce- 
ment. The first four columns of Table 4.2 shows measured strength properties 
of concrete and steel. 



33 



Load P 



! 

Oi 


// 

~1 




\\ 




P/ 2 t fP/ 2 



Figure 4.1 - Typical Cross Section of Circular Pipe. 



Q 
I! 
> 

IT3 
Q 




Figure 4.2 - Finite Element Model of Circular Pise, 



34 



TABLE 4.1 - Geometric Characteristics of the Analyzed 
Pipes to Compare Test and CANDE Results 



Pipes 


Di 


ti 


Asi 


Aso 


tbi 


tbo 


a 




(in) 


(in) 


(in 2 /in) 


(in 2 /in) 


(in) 


(in) 


(in) 


J 


48 


5 


.01683(L) 


. 01233 (L) 


1.10 


1.09 


2 


K 


48 


5 


.02708(H) 


.01992(H) 


1.13 


1.11 


2 


B 


72 


7 


. 03142 (L) 


. 02342 (L) 


1.14 


1.12 


3 


* 
G 


72 


7 


. 05158 (M) 


. 03692 (M) 


1.18 


1.15 


3 


D 


72 


7 


.07292(H) 


.05158(H) 


1.21 


1.18 


3 


* 

Q 


108 


9 


.05158(L) 


. 04000 (L) 


1.18 


1.16 


4.5 


p 


108 


9 


.10317(H) 


.07383(H) 


1.18 


1.15 


4.5 



They have stirrup reinforcement 



1 in = 2.54 cm 



TABLE 4.2 - Material Properties Obtained From 
Tests (15) for the Pipes to be Analyzed 



Pipes 


f 
c 


ft 


* 

fsu 


fsy 


V 
c 


■k* 

fs 




(psi) 


(psi) 


(psi) 


(psi) 


(psi) 


(psi) 




(cylinder) 








average 


average 


1 
J 


4470 


- 


81800 


79250 


4600 


80525 


2 


4730 


- 










1 


4900 


503 


79400 


77000 


5225 


78200 


K 














2 


5550 


465 










1 


4900 


- 


87300 


82000 


4400 


84650 


B 2 


4120 


- 










3 


4640 


- 










4 


3950 


- 










1 


4765 


- 


88650 


85000 


4760 


86825 


G 2 


4375 


- 










3 


5136 


- 










' \ 


6090 


— 


86100 


81500 


5820 


83800 


5550 


— 










1 


5085 


507 


79100 


75500 


5810 


77300 


Q 2 


6540 


578 










1 


5175 


555 


87325 


85000 


5095 


86160 


P 2 


5015 


568 











The average from the inner and outer reinforcement 
** 

The average between the ultimate and yielding stresses 



1 psi = 6.895 kPa 



36 



4.3 ANALYTICAL MODEL AND COMPARISON OF RESULTS 

The circular pipe is idealized using the finite element model 
shown in Figure 4.2 composed of eleven beam- rod elements. For each 
of the seven pipes selected there are two or more test results using 
the same pipe with the same amount of reinforcement, where some of the 
material properties were obtained for each repeated test as shown in 
Table 4.2. For analytical predictions, concrete compressive strengths 
f from repeated tests are averaged. The value of the steel yield 
stress used for analysis is taken as the average between the ultimate 
and yielding stresses obtained from the tests. Averaging the ultimate 
and yield stress of the reinforcement permits considering both ultimate 
load as well as the load-deflection curve within the limits of perfect 
plasticity. The last two columns in Table 4.2 show average strength 
values for concrete and steel used for analysis. 

Except for the cracking strain parameter e , the remaining material 
parameters for steel and concrete are assigned the standard values 
(Table 3.1 and 3.2). Since cracking strain is a sensitive parameter 
and not well established from the test data, two values are assumed 
for analysis; 0.00003 and 0.00008, under the assumption that actual 
values will be within this range. 

In the following, the analytical predictions (CANDE) are compared 
with experimental results for load-deformation, cracking load, and 
ultimate load. 

Load-Deformation . Figures 4.3 to 4.16 show predicted and measured 
vertical and horizontal deflections versus the applied load for each of 
the seven pipes. Each plot shows at least two "repeated" experimental 
tests, two predicted curves representing e = 0.00003 and 0.00008, and 
the actual mode of failure; flexural or shear. Overall it is observed, 
the CANDE predictions generally bracket the experimental curves and 
follow the deformation trends quite well. Results are generally in 
better agreement when ultimate failure is in flexure rather than shear. 
For shear failures, the predicted deflections are generally less than 



37 



A Load P (Kips) 



GO- 




CANOE (5^.00008) 

• CANOE (E t =;00003) 
sb. tests 
(flexural failure) 

0.8 
Vertical Oef lection (in) 



Figure 4.3 - Vertical Load - Vertical Deflection of Pipe J. 



A Load P (Kips) 




CANOE (£ t =.0000S) 

CANOE (8t=.00003) 



lab, tests 



^— — ^ 



0.2 



(ftexural failure) 



0.4 0.S 0.8 

Horizontal Deflection (in) 



Figure 4.4 - Vertical Load - Horizontal Deflection of Pipe J. 



38 



A Load P(Kips) 




02 



QA 0.6 



CANOE (6f».C0008) 

CANOE (E t -00003) 

. lab. tests 

(diag. tension failure) 



0-8 
Vortical Oeflection (in) 



Figure 4.5 - Vertical Load - Vertical Deflection of Pipe K. 



A Load P (Kips) 



60- 
50- 

40' 

30 

20 

10 




CANOE (6 t =.CCC08) 

CANOE (S+-.00C03 ) 

lab. tests 

(diag. tension failure) 



OjS m OS 

Horizontal Deflection (in) 



Figure 4.6 - Vertical Load - Horizontal Deflection of Pipe Z. 



39 



a Load P (Kips) 




0.2 



CAN0£(6t = « 00OO8) 
CAN0E(8t=.0O0O3) 
ab. tests 
(diag- tension failure) 



o,4 as as 10 

Vertical Deflection (in) 



Figure 4,7 - Vertical Load - Vertical Deflection of Pipe B. 



A Load P (Kips 




- CANOE (5t=.00C03) 

- CANOE (6 t =.00003) 
ib. tests 



(diag. tension failure) 



0.8 1.0 

Horizontal Deflection (in) 



Figure 4.8 - Vertical Load - Horizontal Deflection of Pise 3. 



40 



A Load P (Kips 




0.2 



n r- 



0.4 



CAN0E(£ t =-C'COOS) 

CAN0£(£t=OCC0o) 

lab. tests 

(flexural failure, stirrups: 



QJS 0-S 1-0 

Vertical (Deflection (in: 



Figure 4.9 - Vertical Load - Vertical Deflection of Pipe G. 



A Load P (Kios) 




CANOE (Sr-CCC0S) 

CANOE (8 t = CCCCo) 

lab. tests 

(flexurai failure, STirrucs) 



o.a o.4 o.s o-Q io" 

Horizontal Ce flection (in] 



Figure 4.10 - Vertical Lead - Horizontal Deflection of Pipe G. 



41 



A Load P(Kips) 



120- 




— CANDE (£t=-00008) 

— CANOE (E t =.00003) 



ab. tests 



(diag. tension failure) 



0.1 



0.2 0.3 0.4 0.5 

Vertical Oeflection (in) 



Figure 4.11 - Vertical Load - Vertical Deflection of Pipe D. 



A Load P ( Kips) 




0.1 



0.2 



0.3 



CANOE (6f =00008) 

CA NO E(8"t =.00003) 

lab. tests 

(diac;. tension failure) 



OA 



05 



Horizontal Oeflection (in) 



Figure 4.12 - Vertical Load - Horizontal Deflection of Pipe D. 



42 



A Load P (Kips) 



140- 
120- 




CANOE (£ t =.OOCCS) 

CANDE(£t=-00003) 

ab. tests 
(flexural failure, stirrups) 

— > 



i-o 

Vertical Deflection (in) 



Figure 4.13 - Vertical Load - Vertical Deflection of Pipe Q. 



^ Load- P (Kips) 



140 




CANOE (£f=CCCCS) 

CANOE S-^CCCCc) 

ab. Tesrs 
(flexural failure, stirrucs; 



0.S 0.8 1.0 

Horizcntal C2fec~.cn [in! 



Figure 4.14 - Vertical Load - Horizontal Deflection of ?ioe Q. 



43 



160 
140 
120 
100 
80 
SO 

40 
20 



A Load P (Kips) 




CANO£(E t =.00008) 

CANOE (6t=O0003) 

lab. tests 



0.2 



04 



(diacj. tension failure) 
— > 



0.6 0.8 

Vertical Oeflection (in) 



Figure 4.15 - Vertical Load -Vertical Deflection, of Pipe P. 

<^Load P(Kips) 




0.2 



0.4 



CANOE (£t--000C8) 

CANOE (Ef=0CC03) 

lab. tests 

(diag. tension failure) 

— i : ' > 

0.6 0-8 



Horizontal Oeflection (in) 

Figure 4.16 - Vertical Load - Horizontal Deflection of Pipe P. 



44 



measured values. This is to be expected since the CANDE model does not 
account for reduced stiffness due to diagonal cracking as shear failure 
develops. 

For design service loads, say to 2/3 ultimate, the predicted 
curves with e = 0,00008 correlate more closely with measured data 
than predicted curves with e = 0.00003. 

Cracking Load . The cracking load in three-edge bearing is defined 
here as the applied load on the test pipe at which a 0.01 inch crack- 
width occurs extending over a foot in length. The so-called D-load for 
0.01 inch cracking (D m ) is the above cracking load per foot of pipe 
length divided by the inside pipe diameter. ASTM C76-66T describes 
five strength classes of reinforced concrete pipe in terms of D . 

Table 4.3 shows D values; as measured from the experiments, as 
predicted from CANDE, and as specified from ASTM C76 for comparable 
steel areas. The CANDE predictions (which employ the Gergely-Lutz 
formula in Chapter 3), are for the case e = 0.00008 which better 
represents the experimental data. Comparing CANDE predictions with 
test results, good agreement is observed overall. The worst case 
occurs for the medium size pipe with heavy reinforcement (pipe D) where 
the CANDE prediction is 30% higher than measured. If e is reduced, 
the CANDE predictions for D are also reduced. 

In view of the random nature of cracking and the inherent approxi- 
mations in the Gergely-Lutz formula the CANDE predictions are considered 
very reasonable. As a point of interest, the predicted steel stress 
was approximately 50 ksi (345,000 kPa) for all pipes when the Gergely- 
Lutz formula predicted 0.01 inch cracking. 

The ASTM D_. values are shown only for reference and indicate 
bracketing values for the steel areas used in the experimental tests. 

Ultimate Loads . Of the seven pipe types considered in- this study, 
three failed in flexure and four in shear. Two of the pipes failing 
in flexure had stirrup reinforcement to prevent shear failure (diagonal 
cracking) at an early load. The modes of failure predicted by CANDE 



45 



TABLE 4.3 - Comparison of D-load at 0.01" Crack 

Between the Test, ASTM and CANDE Results 



PIPES 


D-Load (0.01" Crack) 


TEST 


TEST 
(average) 


ASTM 
(range) 


CANDE 


(a) 
J 
2 


1190 
1250 


1220 


1350* 


1145 


(b) . 

K 1 

2 


1810 


1810 


1350 - 2000 


1771 












(b) x 

B 2 
3 

4 


1040 
1250 
1460 
1835 


1396 


1350* 


1400 


(c) x 

G 2 
3 


2000 
1670 
1670 


1780 


1350 - 2000 


2130 


(d) 

D i 

2 


2292 
2175 


2234 


2000 - 3000 


2972 












(e) 
Q 2 


1650 
1620 


1635 


1350* 


1216 


P l 

2 


2540 
1980 


2260 


1350 - 2000 


2490 



* The asterisk implies the D value is less than the minimum 
ASTM rating (1350). 



46 



agreed with observed failure modes when stirrup reinforcement was taken 
into account. 

Table 4.4 shows the comparison between test results and CANDE 
predictions for the applied load at ultimate flexural failure. CANDE 
predictions are in excellent agreement with test data. Predicted ultimate 
loads in flexure occur when the slopes of the load-deflection curves be- 
come flat, indicating a collapse mechanism has formed. 

Table 4.5 shows the load comparison for shear failures. Here, 
three CANDE predictions are shown based on three empirical formulas to 
estimate ultimate shear stress; (1) ACI formula for straight members 
(Chapter 3), (b) Theoretical Modification of Committee 326 for pipes (16), 
and (c) MIT Correlation Test formula for pipes (16), When the maximum 
shear stress predicted by CANDE reaches the value of these empirical 
formulas, shear failure is predicted (assuming no stirrup reinforcement). 
Of the three predictions, the standard ACI formula correlates best with 
experimental data (except for pipe B). 

Summarizing this chapter, we conclude the beam- rod element is per- 
forming very well and is capable of predicting the structural responses 
of concrete pipe throughout the entire loading history. 



47 



Table 4.4. - Flexural Failure of Circular Pipes 



PIPES 


LOAD ** 
(kips) 


LOAD (kips) 

(average) 


LOAD (kips) 
(CANDE) ~ 


J l 
2 


41.6 
41.2 


41.5 


40.8 


1 

G 2 
3 


106.0 
106.0 
106.0 


106.0 


115.2 


k 

Q X 
2 


152.2 
152.2 


152.2 


144.0 



* with stirrups 



** test load on 4 foot pipe lengths 



1 kip = 4.48 kN 



Table 4.5. - Shear Failure Loads 



PIPES 


TEST LOAD (kips) 


CANDE LOAD (kips) 


each 


average 


* 
(a) 


(b) 


(c) 


1 
K 

2 


59.0 
60.2 


59.6 


56.3 


45.6 


52.8 


1 

B 2 
3 

4 


62.4 

64.2 
64.0 
61.5 


63.0 


73.3 


60.0 


67.2 


D 

2 


65.2 
87.2 


76.2 


78.5 


72.0 


91.2 


1 
P 

2 


142.8 
151.2 


147.0 


154.2 


129.6 


165.6 



(a) v = 



2.0 TV 
c 



(Standard ACI) 



(b) V = 1.6 TV + 64 A ,/D. (Committee 326) 

c c si 1 

(c) v - 1.53 JV + 320 A ./D. (MIT Correlation) 

C C SI 1 



48 



CHAPTER 5 

EVALUATION OF REINFORCED CONCRETE MODEL FOR 
BOX CULVERTS LOADED IN FOUR-EDGE BEARING 

• Like the previous chapter, this -chapter continues to examine the 
validity of CANDE's reinforced concrete, beam-rod element. Here, we 
compare CANDE results with experimental data for box culverts tested in 
four-edge bearing. The results to be compared include the load for 
0.01 inch cracking and ultimate load. The experimental data did not 
include load-deformation histories, consequently these cannot be com- 
pared. For reference, the comparisons also include the SGH analytical 
predictions (12) based on an elastic analysis discussed in Chapter 2. 

5.1 EXPERIMENTAL TEST 

The experimental results used for the comparison belong to a test 
program (13,14) where out-of-ground reinforced concrete box culverts 
with welded wire fabric were loaded up to failure. The loading was 
applied as shown in Figure 5,1 using a 4-edge bearing testing apparatus. 
The material properties of concrete were determined using cylinder tests 
and core tests for each kind of box, and the mechanical properties of 
the reinforcement were determined by tensile tests (19). Three span 
sizes of box were tested, small, medium and large with three levels of 
reinforcement in each size, low, medium and high. Thus, nine types of 
boxes were tested with two repeated tests per box type. 

Tables 5.1 and 5.2 together with Figure 5.1 show the measured 

geometries and material strengths for each test box where repeated boxes 

are labelled A and B. Note that the core tests for f ' are generally 

c 

higher than cylinder tests for f and the ultimate steel stress is 10 to 
20% higher than initial yield stress. 

The test program was performed to verify the SGH analysis/design 
method (12) which in turn was used to develop the ASTM standard designs 
for reinforced concrete box culverts (21), 



49 



i 
u 

CSL 



t- 



P/2 P/2 



1 tb2 HVt 



Li 



ASf 



1*3 



IK 



t \ 



•AS4 



--tbi 



t fp-r As 3 



P+W 

2 2 



Figure 5.1 - Typical Cross Section of Concrete Bos Culvert. 



u a „ 
•f — f 



f2 



HH+^ 



R+t 



• • • 



r t 2 3 4 5 
6 
7t 
8* 



| Thv+s 



9* 



*d5 .14 



J77777 



>Vz 



10" -t 

13 12 it l HV+t £ 



1, S/? + t/2~ 
t^ 1 



Figure 5.2 - Finite Element Model of Concrete Box Culvert. 



50 



TABLE 5.1 



Geometric Characteristics of Test Box Culverts 



*•* 

BOXES 


SPAN 
S(in) 


RISE 
R(in) 


HH,HV 
(in) 


(in) 


t3 
(in) 


tb 
(in? 


tt>3 
(in) 


a 
:in) 




















8*4-8 


A 
B 


96 
96 


48 
48 


8 
8 


8.125 
8.250 


8.125 
8.000 


1.376 
1.126 


1.251 
1.251 


12 
12 


8*4-2 


A 
B 


96 
96 


48 
48 


8 
8 


8.188 
8.253 


8.251 
8.188 


1.526 
1.278 


1.214 
1.401 


12 
12 


8*4-18 


A 
B 


96 
96 


48 
48 


8 
8 


8.250 
8.250 


8.251 
8.126 


1.229 
1.416 


1.354 
1.229 


12 
12 


6*4-10 


A 
B 


72 
72 


48 
48 


7 
7 


7.313 
7.313 


7.251 
7.313 


1.053 
1.240 


1.303 
1.240 


9 
9 


6*4-2 


A 
B 


72 
72 


48 
48 


7 
7 


7.375 
7.375 


7.438 
7.438 


1.068 
1.443 


1.006 
1.006 


9 
9 


6*4-22 


A 
B 


72 
72 


48 
48 


7 
7 


7.313 
7.438 


7.375 
7.375 


1.086 
1.523 


1.148 
1.148 


9 
9 


4*4-4 


A 
B 


48 
48 


48 
48 


5 
5 


5.187 
5.188 


5.375 
5.250 


1.353 
1.385 


1.103 
1.353 


6 
6 


4*4-18 


A 
B 


48 
48 


48 
48 


5 
5 


5.188 
5.313 


5.188 
5.188 


1.237 
1.112 


0.862 
1.175 


6 
6 


4*4-2 


A 
B 


48 
48 


48 
48 


5 
5 


5.563 
5.376 


5.188 
5.250 


1.756 
1.694 


1.381 
1.173 


6 
6 



t 9> t_ = thickness of top and bottom slabs respectively 

** Boxes are identified according to Reference (14): i.e., span(f t)*rise(ft) 
design earth cover (ft) for interstate live load. 

1 in = 2.54 cm 



51 



TABLE 5.2 Reinforcement and Material Properties 
of the Box Culverts 



BOXES 


si 
(in2/in) 


A s2 
(in2/in) 


s3 
(in 2 /in) 


(in) 


f c (psi) 
cylinder 


f c (psi) 
cores 


* 

fy 

(ksi) 


fsu 
(ksi) 




















8*4-8 


A 
B 


.02492 


.02492 


.02492 


2 


4934 
4757 


12510 
5515 


72.3 


83.65 


8*4-2 


A 

B 


.04325 


.03550 


.03550 


2 


5288 
4952 


4475 
5425 


82.0 


89.58 


8*4-18 


A 

B 


.04325 


.04325 


.04325 


2 


5111 
5430 


5285 
5855 


80.5 


94.95 






















6*4-10 


A 

B 


.01450 


.02075 


.02075 


2 


6296 
5022 


7060 
7460 


85.9 


94.40 


6*4-2 


A 
B 


.03550 


.03475 


.02675 


2 


5624 
5589 


6680 
6965 


85.8 


99.43 


6*4-22 


A 
B 


.02358 


.03442 


.03442 


2 


5553 
5341 


5960 
7190 


86.2 


95.16 






















4*4-4 


A 
B 


.01117 


.01117 


.01117 


3 


5518 
7534 


6030 
6670 


78.5 


95.80 


4*4-18 


A 
B 


.01117 


.01967 


.01967 


2 


7428 
7729 


7000 
6635 


77.3 


90.93 


4*4-2 


A 
B 


.01600 


.02692 


.02692 


2 


5872 
6155 


5715 
6430 


82.0 


92.73 



*This value is an average of the three reinforcements 
S = spacing of longitudinal wires 

NOTE: A . steel not used 
s4 



1 in = 2.54 cm 
1 psi = 6.895 kPa 



52 



5.2 CANDE MODEL 

Figure 5.2 shows the finite element model for a typical box culvert 
test. Because of symmetry only half the box is modeled with 14 beam- rod 
elements. The element pattern shown was found to be sufficiently accurate 
with regard to element lengths. The reaction support (shown at node 14) 
is modeled with a triangular element rather than a boundary condition in 
order to avoid imposing a moment constraint (a quirk of CANDE) . 

Each element cross section is assigned the concrete thickness, 
the steel area, and steel area locations as actually reported from the 
experiments (Tables 5.1 and 5.2). Haunches at the box corners are 
modeled with two corner elements whose thicknesses are increased by 
one-half the haunch dimensions. 

For concrete material properties, f* is taken from the core tests 
(Table 5.2, except first box) as this is generally more representative 
of each test box, than cylinder tests. The cracking strain is assumed 
as e = 0.0001 for all box tests based on observing typical concrete 
test results. Other concrete parameters are assigned standard values 
(Table 3.1). 

Steel "yield" stress for the elastic-perfectly plastic model is 

taken as the ultimate stress reported in the last column of Table 5.2 

for each box. Ultimate steel stress, rather than initial yield stress 

is assumed because this better approximates ultimate load capacity. 

Other steel parameters are assigned standard values (Table 3.2). Within 

each test box the steel stress-strain properties for A ,, A _, and A „ 

si sZ sj 

are assumed identical. 

5.3 COMPARISON OF MODELS WITH EXPERIMENTS 

In the following comparisons for cracking load and ultimate load 
(flexure and shear), the "load" refers to the total applied load P per 
foot length of test pipe (see Figure 5.1). Each repeated experimental 
test is also repeated analytically with the associated variations in 
geometry and material properties. 



53 



Cracking Load . Table 5.3 shows a comparison between test data and 
CANDE predictions for the load producing a 0.01 inch crack. These cracks 
occur near the centerline on the inside surfaces of the top or bottom 
box slabs. The table specifys "top" or "bottom" indicating which slab 
the 0.01- inch crack was first observed, and the CANDE prediction cor- 
responds to that location. Also shown in Table 5.3, are the SGH pre- 
dictions for cracking loads to serve as a reference. 

Overall it is observed the CANDE predictions are very good and 
are statistically better than the SGH predictions as shown at the bottom 
of Table 5.3, A graphical comparison of the data is shown in Figure 5.3 
from where it is seen that CANDE cracking load predictions are slightly 
lower in the average* (conservative) than the test data, but only on the 
order of 5 to 10%, 

As previously discussed, CANDE predictions are semi -empirical and 
employ the Gergely-Lutz crackwidth formula. Although not reported here, 
the ACI crackwidth formula (4) was tried with CANDE but not found satis- 
factory in this study. 

Ultimate Loads . In loading the 9 pairs of boxes (18 tests) to 
ultimate, 10 tests failed in flexure and 8 tests failed in shear 
(diagonal cracking). Two pairs of boxes produced a failure of each 
kind. Modes of failure predicted by CANDE agreed with observed failure 
modes . 

Table 5.4 shows the comparison of the ultimate load for flexural 
failure between CANDE prediction, the test results, and the SGH analy- 
tical results. The values calculated by CANDE are in very good agree- 
ment with test results, and are slightly better than the SGH analytical 
results. Figure 5.4 shows graphically the comparison between CANDE and 
the test results for ultimate load at flexural failure. CANDE' s overall 
results correlate excellently with the test results, with a + 6% error 
range. 

Table 5.5 shows the comparison of the ultimate load for shear 
failure (diagonal cracking) between CANDE prediction, the test results, 



TABLE 5.3 — Comparison of CANDE Results with Test and 
SGH Results for 0.01 inch Cracking Load 









p . 


01 (lb/f 


t) 


BOXES 


TEST 


CALCULATED 


SGH 

Report (14) 


CANDE 












8*4-8 


A 
B 


Top 
Top 


9250 
11300 


7840 
9430 


9400 
9200 


8*4-2 


A 
B 


Bottom 
Top 


14000 
12300 


10950 
12360 


12300 
12300 


8*4-18 


A 
B 


Top 
Bottom 


13000 
13500 


15650 
13442 


14700 
14200 














6*4-10 


A 
B 


Bottom 
Bottom 


9500 
9500 


5830 
6220 


7200 
7500 


6*4-2 


A 
B 


Bottom 
Bottom 


14500 
10500 


10640 
10650 


10900 
10800 


6*4-22 


A 
B 


Bottom 
Top 


15000 
12500 


12510 
11090 


12000 
11000 














4*4-4 


A 
B 


Bottom 
Top 


6700 
6000 


2740 
2700 


3600 
4200 


4*4-18 


A 
B 


Bottom 
Top 


7000 
8000 


7770 
7090 


7800 
7500 


4*4-2 


A 
B 


Bottom 
Bottom 


7800 
8500 


6940 
8380 


7500 
8600 










p .oi test 


Average 


1.29 


1.10 


Standard 
Deviation 


0.43 


0.24 


P m calc. 


. Ul 






Coefficient 
of Variation 


34% 


21% 



1 lb/ft = 14.6 N/m 
55 



16 
14 

P.01 test 12 
( K/f t) 




+ 8x4 Boxes 
• 6x4 Boxes 
x 4x4 Boxes 



8 10 12 14 16 

P.01 calculated 
(K/ft) 



P.01 (test) 



■ = 1.1063 



P.01 (calculated) 



Roi (test) 



P.01 (calculated) 



= 1.0475 



* Excluding O 



Figure 5.3 - Comparison of Test and Calculated 0.01 inch 
Crack Load. 



56 



TABLE 5.4 - Comparison of CANDE Ultimate Load with Test and 
SGH Ultimate Loads for Flexural Failure 



BOXES 


uf 


(lb/ft) 




TEST 


CALCULATED 


SGH 

Report (14) 


CANDE 










' 8*4-8 


A 

B 


17860 
17230 


16050 
15780 


17700 
16800 


8*4-2 


A 
B 


29690 


28200 


27600 


8*4-18 


A 

B 


















6*4-10 


A 
B 


16100 
15000 


15380 
15390 


15600 
15600 


6*4-2 


A 
B 








6*4-22 


A 
B 


















4*4-4 


A 
B 


8980 
8440 


9800 
9011 


9600 
9600 


4*4-18 


A 
B 


13150 
13170 


12680 
12730 


13200 
13200 


4*4-2 


A 
B 


19300 


18510 


18000 










P _ test 
uf 


• 


Average 


1.03 


1,01 


Standard 
Derivation 


0.06 


0.06 


P calc. 
uf 


Coefficient 
of Variation 


6% 


. 5.8% 



1 lb/ft 



14.6 N/m 



57 



30 




+ 8x4 Boxes 
•• 6x4 Soxes 

x 4x4 Boxes 



10 12 14 



16 18 20 22 24 26 28 30 
Pllf calculated 



(K/f t) 



^PUf (test) 
2PUf (calculated) 



1.0129 



:P0f (test: 



=.9897 



'PUf (calculated) 






4fr Excluding 
Figure 5.4 - Comparison of Test and Calculated Ultimata Flexural Load. 



58 



TABLE 5.5 - Comparison of CANDE Ultimate Load with Test 
and SGH Ultimate Loads for Shear Failure 







Pu dt (lb/ft) 


BOXES 


TEST 


CALCULATED 


SGH 
Report (14) 


CANDE 


8*4-8 


A 
B 








8*4-2 


A 
B 


22520 


21520 


22000 


8*4-18 


A 
B 


20890 
24490 


21590 
22860 


21600 
23000 












6*4-10 


A 
B 








6*4-2 


A 

B 


19400 
25250 


23420 
23950 


22800 
24000 


6*4-22 


A 
B 


25'680 
21150 


21260 
23530 


21600 
23400 












4*4-4 


A 
B 








4*4-18 


A 
B 








4*4-2 


A 
B 


14080 


12790 


13200 










Pu, 

dt test 


Average 


1.02 


1.01 


Standard 
Deviation 


0.12 


0.10 


p. , 


dt cal 


c. 


Coefficient 
of Variation 


12% 


10% 



1 lb/ft = 14.6 N/m 



59 



30 






28- 






26- 






Pu dt test 24- 




+» / 


(K/ft) 22 . 

20- 






18- 






16- 






14- 


x/ 




12- 




-h 8x4 Boxes 
•■ 6x4 Boxgs 


10- 




x 4x4 Boxes 



10 12 



8 20 22 24 26 28 30 
PlU+ calculated 



'dt 



(K/ff) 



Pbdt ^est) 



Pu^t (calculated) 



•= 1.0108 



Figure 5.5 - Comparison of Test and Calculated Ultimate Diagonal 
Tension Load. 



60 



and the SGH analytical results. The values obtained from CANDE are 

assuming that the maximum shear stress resisted by the concrete is 

2.0 / f f . Once again the values obtained from CANDE are very close to 
c 

the test results and are a better prediction than the analytical results 
of SGH. The test results are compared graphically with CANDE results 
in Figure 5.5, from where we can observe that the amount of error from 
CANDE is in the range of + 10%, a very good correlation for practical 
purposes. 

The performance of the beam-rod element used in the CANDE program 
to model reinforced concrete box culverts has been shown to perform 
very well in out-of-ground loading. Subsequent studies will consider 
the box culvert buried, subjected to soil loads as well as live loads. 
The empirical formulas for crack prediction and shear resistance used 
in CANDE will be the same ones used in this chapter, where their per- 
formance was found satisfactory. 



61 



CHAPTER 6 

DEVELOPMENT OF LEVEL 2 BOX MESH 

The reinforced concrete box culvert model is now considered for 
its actual function as a conduit buried in soil. Accordingly, both the 
soil and the box form the structural system, hereafter called box-soil 
structure. The soil plays a dual role; on the beneficial side it adds 
substantial stiffness to the box-soil structures, on the detrimental 
side it transits gravity and applied loads to the box during the in- 
stallation process. 

To determine loads acting on the box requires a complete model of 
the box-soil structure simulating the entire installation process. In 
this chapter a general finite element model of the box-soil structure 
is presented with the intent of developing an automated finite element 
mesh subroutine suitable for simulating the vast majority of box-soil 
installations encountered in practice. This is called the level 2 box 
option of CANDE. 

To develop an automated finite element mesh requires some limiting 
assumptions and specifications of a variety of parameters describing 
the box-soil system. Overall assumptions are symmetry about the ver- 
tical centerline and plane strain geometry and loading. Adjustable 
system parameters include; box dimensions (span and rise), soil boundary 
dimensions (width from centerline, depth below box, and height of cover 
above box), soil zones (in situ soil, fill soil, and bedding soil) and 
installation type (embankment or trench). These parameters along with 
the question of mesh refinement are discussed in the following. 

6.1 PARAMETERS TO DEFINE THE MODELS 

The depth and width of the entire soil zone are specified in 
terms of the particular box dimensions being analyzed. The box culvert 



62 



is idealized with beam elements located along the middle line of the 
walls, so the nominal box span used in our model is equal to the inside 
span of the box plus its thickness. Likewise for the nominal rise of 
the box. Defining Rl as half the nominal span and R2 as half the 
nominal rise, as shown in Figure 6.1, the soil depth below the box is 
set at 3R2, and the soil width is set at 4R1 from the box sides. These 
soil boundaries are adjudged to be outside the zone of soil-structure 
interaction based on previous studies (1). 

The height of cover (see Figure 6.1) is an input parameter denoting 
the final fill height above the box. However, the height of the mesh 
over the top of the box is limited to 3R2 or the specified height of 
soil cover, whichever is less. For cover heights greater than 3R2 
equivalent loading is used as discussed subsequently. 

Other geometry parameters that need to be defined are the trench 
depth and trench width as shown in Figure 6.1. If the mesh model is 
intended to represent an embankment installation, the trench width is 
4R1 so that only fill soil exists on the sides and in situ soil is 
leveled with bottom of the box. 

The material zones are in situ soil, bedding and fill soil, where 
each zone can be assigned the same or different soil mechanical pro- 
perties. In addition to Rl and R2, the box culvert geometry is defined 
with the side, bottom and top slabs thicknesses, and haunch dimensions 
as shown in Figure 6.2. The amount of steel reinforcement around the 
box is defined by steel areas A , A , A and A along with a common 
cover thickness as shown in Figure 6.3. 

So far only the parameters of the box-soil systems and the general 
dimensions have been discussed, nothing has been said about the finite 
element mesh itself or the sequence of loading. The example shown in 
Figure 6.4 will be used to explain the mesh arrangement. All the di- 
mensions and height of soil cover are shown. A trench configuration 
is used with three zones of soil for the system. The values of Rl and 
R2 give the overall size of the mesh as previously discussed (see 



63 



A 




CM 

cr 



00 



4 R1 



Figure 6.1 - Parameters to Define Buried Concrete Box Culvert. 



64 



Z R2 




i- 



-* ' *-PTS 



Figure 6.2 - Parameters to Define the Geometry of the Bos Culvert, 




AS4 



XL1-- kL 

XU "R1 



Figure. 6.3 - Parameters to Define the Reinforcement of the Box Culvert. 



65 



3R2 



3R2 




4 R1 



Figure 6.4 - Example of Box Culvert to Define a Mesh. 



66 



Figure 6.1). The finite element mesh configuration for the example is 
shown in Figure 6.5, where by symmetry only half of the box-soil system 
is modeled. The same figure shows the soil elements and the nodal points 
of the mesh. The number of layers of soil elements on top of the box 
can decrease if the height cover of soil is less than 3R2, but will 
never be less than two rows. The soil elements near the box are smaller, 
so that a more refined mesh around the box can provide a better behavior 
of the box-soil model where the stress gradients are known to be highest. 
The soil elements are four node, nonconforming quadrilaterals with ex- 
cellent performance characteristics (1). The coordinates of the nodes 
are all related to Rl and R2 and, if desired, can be changed using the 
extended level 2 option (see Appendix B). 

The loading sequence, called incremental construction (1,2), simu- 
lates the actual installation process of placing soil layers in a series 
of lifts. Figure 6.6 shows the construction increment numbers of element 
groups entering sequentially into the system. The initial system (first 
construction increment) includes all in situ soil, bedding, and the box 
loaded with its own body weight. Subsequent increments, numbers 2 
through 9, are gravity loaded layers of fill soil. For specified heights 
of soil cover less than 3R2, the mesh over the top of the box is assigned 
proportionally less soil layers. If the height of soil cover is greater 
than 3R2, the load due to the soil over 3R2 is applied as equivalent 
overburden pressure increments. This load sequence is applied after 
the ninth soil layer using n-9 additional load increments, where n is 
the total number of construction increments specified in the input. 

The box-soil mesh described here is generated automatically using 
the CANBOX subroutine to generate all the necessary data required to 
define the finite element mesh of the system. CANBOX subroutine para- 
meters, options and mesh size are discussed with more detail in Appendix 
B. 

6.2 ASSUMPTIONS AND LIMITATIONS 

When using this automatic mesh generation, there are some assump- 
tions involved that should be remembered. 



67 



MOCfiL MUM3£?.S 



137 



146 



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147 



159 



150 



I SI 



152 



143 143 ISC iSl 



153 



152 



154 



153 



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155 



ISS 



135 



135 



137 133 



124 



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113 



114 



125 



113 



139 



127 



123 



140 



129 



141 



142 



130 



131 



115 



102 103 104 105 



So 57 S3 39 



4S 



46 47 



117 



108 



113 



113 



120 



107 



103 



109 



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95 



98 



39 



81 



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67 



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43 43 



97 



93 



90 



91 



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S4 



77 



63 



r 



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70 



SO SI 



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143 



132 



121 



110 



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23 



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13 14 IS 15 



17 



30 



13 



13 



12 3 4 3 5 



31 



20 



32 



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10 



11 



Figure 6,5 - Under ormed Grid with Nodal Points. 



68 



INCREMENT NUneERS 



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Figure 6.6 - Load Increment: Pattern for the Layers of Soil. 



69 



a) To simplify the mesh, we are assuming symmetry about the 
vertical axis so only half of the box-soil system is analyzed. This 
assumption implies that only symmetric loading can be applied to the 
box when using concentrated loads on top of the mesh. For most of 
the cases a symmetric loading arrangement satisfies the loading cond- 
ditions . 

b) Three different zones of soil can be specified within the 
soil mesh; fill soil, bedding soil, and in situ soil. 

c) When a trench condition is specified, the in situ soil 
forming the trench extrados goes all the way up to the top of the 
mesh, whereas for the embankment condition, the in situ soil remains 
at the level of the bottom slab of the box culvert. 

d) For the concrete box culvert, the thickness of the wall is 
constant along a particular side but may vary between sides. However, 
the concrete cover of the reinforcement is the same for all sides. 

Many of the above assumptions can be removed by use of extended 
level 2 option (see Appendix B ) which allows selective modification 
of the automated mesh discussed above. Virtually all limitations can 
be removed by use of level 3 option wherein the user defines his own 
mesh (1,2). 



70 



CHAPTER 7 

EVALUATION OF CANDE BOX-SOIL SYSTEM 

In the previous chapters the reinforced concrete beam-rod element 
was evaluated with experimental data for out-of-ground structures from 
which we concluded that the beam-rod element itself performs satisfact- 
orily. In this chapter we examine the performance of the reinforced 
concrete model as a buried box culvert, where the soil-box structure 
is modeled with the level 2 box finite element idealization described in 
the previous chapter. Using this structural system the loads acting on 
the box are not prespecified, but rather are determined from the finite 
element solution of the box-soil system. Thus, the performance of the 
box culvert model depends, in part, on the responses of the soil system. 

To evaluate the CANDE box-soil model we first consider a parameter 
sensitivity study to assess the influence of soil stiffness and installation 
type on the structural behavior of a typical box culvert. Secondly, we 
compare the CANDE predictions with full scale field test data (24), pro- 
viding a direct validation of the box-soil model. 

7.1 SENSITIVITY OF SOIL PARAMETERS 

In this section the influence of soil parameters on the structural 
performance of a particular box section is examined. Soil parameters 
considered include; elastic properties and type of installation (trench 
or embankment). 

The particular box section (hereafter called standard box) used to 
examine sensitivity of the soil parameters was obtained from the ASTM 
Standards (21) for box sections under earth dead load conditions. An 
intermediate size box with medium reinforcement was chosen to be repre- 
sentative for this study. Specifically, the standard box has 8 feet (2.4 m) 



71 



span, 6 feet (1.8 m) rise and 8 inches (20.3 cm) wall thicknesses (8*6-8) 
with a specified design earth cover of 10 feet (3.05 m) . The material 
properties for the standard box culvert are: 



f 
c 



and 



TC 
S„ 



= 5000 psi 
(3A500 kPa) 

= 0.0001 



= 65000 psi 
(448000 kPa) 

= 150 pcf 

(23.5 kN/m3) 

= 4286.8 ksi 
(29550 MPa) 

= 0.017 



= 29000 ksi 
(200000 MPa) 

= 0.30 



1.25 inch 
(3.18 cm) 

2.0 inch 
(5.08 cm) 



(unconfined compressive stress of concrete) 
(maximum tensile strain of concrete) 
(yield stress of reinforcement) 
(unit weight of concrete) 
(concrete Young's modulus) 
(concrete Poisson's ratio) 
(steel Young's modulus) 
(steel Poisson's ratio) 

(concrete cover) 

(spacing longitudinal reinforcement) 



The characteristics of the 8*6-8 box cross section are shown in 
Table 7.1, where the nomenclature is referred to the typical cross section 
shown in Figure 7.1. 

For the purposes of this study, elastic soil properties are assumed 
in a range covering stiff, medium and soft soils, where their parameters 
are Young's modulus and Poisson's ratio. Table 7.2 summarizes the pro- 
perties of the in situ, bedding and fill soil for the soft, medium and 
stiff soil model. The fill soil weight density is assumed 120 pcf (18.8 
kN/m-*) for all types so that only stiffness is varied. Bedding and 
in situ soil zones are not assigned a weight density since they form the 
initial configuration. 

Installation type . For the first study the influence of installation 
type on the standard box is considered for a trench condition versus 



72 



CM 

<r 

CM 




Figure 7.1 - Typical Cross Section and Parameters co Define a 
Concrete Box Culvert. 




Figure 7.2 - Definition of Vertical Deflection o and Horizontal 



Deflection 5, 



v 



a* 



73 



TABLE 7.1 Characteristics of Standard Box Culvert 
Used in Sensitivity Study 



Box 
(ft*ft-in) 


ASTM Design 
Earth Cover 
(ft) . 


si 
(in2/in) 


s2 
(in 2 /in) 


s3 
(in2/in) 


A A 
s4 

(in 2 /in) 


XL1 


PT 
(in) 


8*6-8 


10.0 


0.01667 


0.02417 


0.02583 


0.01583 


0.50 


8.0 



From Table 3 of ASTM Standards (21). 



TABLE 7.2 Properties of the Linear Soil Models 
Used in CANDE Solution 



Type 


* 

of Soil 


Young ' s Modulus 
(psi) 


Poission's 


Ratio 




Insitu (1) 


333 


0.33 




SOFT 


Bedding (2) 


666 


0.33 






Fill (3) 


333 


0.33 






Insitu (1) 


2000 


0.33 




MEDIUM 


Bedding (2) 


4000 


0.33 






Fill (3) 


2000 


0.33 






Insitu (1) 


3333 


0.33 




STIFF 


Bedding (2) 


6666 


0.33 






Fill (3) 


3333 


0.33 





Unit weight is 120 pcf for fill soil 

1 ft = 0.3048 m 

1 in = 2.54 cm 

1 psi = 6.895 kPa 

1 pcf = 157.1 N/m" 



74 



an embankment condition. The trench width beyond the box sides is taken 
as 2.0 feet (0.61 m) (narrow trench) and soil properties for both instal- 
lation types are assigned medium stiffness values (see Table 7.2). In 
both cases, the box was loaded up to 28 feet (8.53 m) of soil cover above 
the box. Figure 7.2 shows the definition of vertical and horizontal 
relative displacements used in subsequent discussions. 

Figure 7.3 shows the fill height versus vertical displacement history 
of the box, from where it's observed that the embankment configuration 
produces slightly greater vertical deflections in the box. Figure 7.4 
shows the bending moment diagrams and shear force diagrams in the box 
at 28 feet of soil cover for both installation types. The embankment 
condition gives greater bending moments and shear forces acting in the 
box, which conforms to the greater deflections previously observed. 

From this comparison it was concluded that the embankment condition 
produces slightly greater loading conditions on the box so that in all 
subsequent studies presented herein only the embankment condition will 
be considered. 

Soil Stiffness . The effect of elastic soil stiffness is investigated 
using the same standard box (8*6-8) with an embankment soil configuration, 
where the properties of the soil are varied to idealize a soft, medium 
and stiff soil as defined in Table 7.2. With these values a range of 
variation is covered so the effect of each can be observed. Figure 7.5 
shows the load versus vertical deflection history of the box for the 
three classes of soil stiffness, where the box is loaded up to failure 
for each case. The failures for the box culverts are defined by exceeding 
ultimate shear capacity (V ) or by the formation of plastic hinge mechanisms 
from excessive moments and thrust (M ) . In this study shear failure 
occurs before plastic hinging in the standard box (8*6-8) for all the 
three types of soil. Failure occurs first for the soft condition, whereas 
for the medium and stiff soil conditions failure occurs at a greater 
height of soil cover. Note that when ultimate bending failure (M ) occurs, 
the deflections do not show a flat slope (increase without bounds) as 



75 



.Height of So 
i (ft) 



25- 



20- 



15 



to 



5 




— Troncn 

— embankment 



8x6-8 Box 

medium reinfor. 
standard soif 



0.1 



02 



03 



0.4 



**fv 



Vertical Oisplacemen 
(in) 



Figure 7.3 - Height of Soil Over the Top of the Box - Vertical 
Deflection for Trench and Embankment Situations. 



76 





(a) Bending Moment diagram 
for trench ■ 2.0 ft. 



(b) Bending Moment diagram 
for embankment 




Cc) Shear Force diagram 
for trench 3 2.0 ft. 




(d) Shear Force diagram 
for embankment 



Figure 7.4 - Bending Moment Diagrams and Shear Fo'rce Diagrams of 
8x6 Box Culvert with 28 ft of Soil Cover for Trench 
* 2.0 ft and. Embankment Conditions. 



77 



Mu 

/ 

A Heiqht of soil / 

* (ft) /***, 

Vu / 



25- 



20 




Mu 



Vu 



Stiff SOil 

medium soil 
soft soil 



8x6-8 Sox 
medium reinforce: 



L— 



0.2 



0.4 



05 



0.8 



■»f 



v 



Vertical Displacement 

(in) 

Figure 7.5 - Height of Soil Over Che Top of che Box - Vertical 
Deflection for Three Kinds of Soil. 



78 



they did in previous studies with out-of-ground culverts with applied 
loads. This is because the soil stiffness is now controlling deflections 
at ultimate. Accordingly, the slope of the deflection curves at ultimate 
is in near proportion to soil stiffness. From Figure 7.5 it is evident 
that the soft soil condition is restricted to smaller cover heights to 
reach ultimate than the stiff er soils. Thus, the medium and stiff soil 
conditions are more favorable for the box behavior. 

Figure 7.6 shows the bending moment diagrams for the box at bending 
failure (presuming stirrups) for the three types of soil conditions. It 
is observed that even when the failure occurs at different heights of 
soil cover, the maximum moments are similar as would be expected for 
ultimate moments. From this comparison it is evident that the type of 
soil is an important factor for analysis and design of a box culvert. 

7.2 COMPARISON WITH TEST DATA 

To validate the box-soil model, CANDE results are compared with test 
data from a full scale field installation. Test data on buried box 
culverts is very limited. However, recent research reports from the 
Department of Transportation, Lexington, Kentucky (23,24,25) have supplied 
some test data. From these reports, data was obtained for a box culvert 
in Clark County, Kentucky, designed as an embankment with a yielding 
foundation within a bedrock formation (24). Instrumentation on the box 
included normal pressure gages and a few strain gages on reinforcement 
steel which were reported not to function properly, thus only normal 
pressure comparisons are used for this study. The box is identified 
as Station 123+95 in the report (24) and its cross section as modeled 
by CANDE is shown in Figure 7.7 along with reinforcement areas. 

The level 2 box embankment condition is used for the model with 
three zones of soil that are assumed linear with the properties shown 
in Table 7.3. The in situ soil is bedrock so a large value is assumed 
for its modulus of elasticity. Table 7.4 shows the material properties 
used in CANDE to model the box culvert. The concrete and steel strengths 
used in CANDE were obtained from data presented in a report (25) and the 



79 




(a) Stiff soil and 31 ft of soil cover 







A 


189*7 




CD 


y 


/ 




CO 


^^ 


/ 




iO — ' 


I 


M 


db-in) 

CO 


\ 






X 


21400 



(b) Medium soil and 28 ft of soil cover 




(c) Soft soil and 22 ft of soil cover 



Figure 7.6 - 3ending Momenc Diagram of 8x6 Box Culvert at 3ending 
Failure Usinc Three Different Kinds of Soil. 



80 




z 










77777777777777777T77/ 

(solid rock)//// f * 

y////////M» 




Tfgure-7.7 Cross Section of Buried Test Box Culvert 
(Station 123+95). 



81 



TABLE 7.3 - 



Linear Soil Properties for Test 
Box Culverts 



SOIL 


Young's Modulus 
(psi) 


Poisson's Ratio 








INSITU 
(rock) 


100000 


0.25 


BEDDING 


4000 


0.25 


FILL 


2000 


0.25 



Note: soil weight density of 138 pcf for fill soil 



TABLE 7.4 - Box Culvert Properties 



BOX 


f 

c 

(psi) 


Y 
c 

(pcf) 


f 

y 

(psi) 


e 
t 


Station 

123+95 


4500 


150 


60O00 


.0001 



1 psi - 6.895 kPa 
1 pcf = 157.1 N/nf 



82 



other properties are assigned standard values (Tables 3.1 and 3.2). 

The box is loaded up to 77 feet (23.5 m) of soil cover using small 
load increments thereby obtaining a history of the box-soil system 
performance. However, the only information that can be used for com- 
parison is the pressure distribution on the box which was experimentally 
measured at two heights of fill soil, 21.6 feet (6.58 m) and 77 feet (23.5 m) . 

To measure the pressure around the box, eight Carlson earth pressure 
cells were installed, two on each side of the box. Figure 7.8 shows the 
CANDE pressure distribution around the box at the two fill heights of 
soil, along with the measured test data. CANDE predictions of the pres- 
sure is very close to the measured value for the top and bottom slabs. 
The measured pressure on the sidewalls is different for the right wall 
and the left wall, and CANDE prediction is closer to the values measured 
for the right wall. 

From CANDE, some interesting observations are, when loaded up to 
the maximum 77 feet (23.5 m) of soil cover, the bottom corner steel 
started to yield for the last load increment. Also, the 0.01 inch 
(0*0254 cm) crack first developed with 60 feet (18.3 m) of soil cover. 
These observations suggest an economical design was achieved with no 
conservatism. 

After this last study it can be said that the reinforced concrete 
beam element to model the box culverts and the box-soil system appear, 
to give reasonably good predictions of the behavior and performance of 
buried box culverts. 



83 



fill heiqht 21.6' 1 ^ AKin r «~ , 

J > CANOE- normal pressure 

fill height 77' J 

instant value (H= 21.6') 

, > test- normal pressure 
lapsed time value (H=77) J 




SCALE: 



50 psi 



Figure 7.8 - Comparison of Test Data with CANDE Prediction 
of Pressure Over the Box Culvert. 



84 



CHAPTER 8 

EVALUATION OF ASTM C-789 DESIGN TABLES WITH CANDE 

In the previous chapter, CANDE' s reinforced concrete beam-rod 
element has been developed and compared with experimental data for both 
in ground and out-of-ground culverts. Overall, very good correlation 
was observed for all aspects of structural performance, including; 
load-deformation curves, cracking loads, ultimate loads, and soil pres- 
sures, thereby lending a measure of confidence and validity to the 
CANDE model. 

In this chapter the objective is to cross-evaluate CANDE with ASTM- 
C789 design tables for buried box culverts (21). As discussed in Chapter 
2, the ASTM design tables are based on an elastic method of standard 
analysis together with the ultimate method of reinforced concrete design 
(12) . However, the magnitude and distribution of loads acting on the 
box are assumed, as opposed to determining loads with soil-structure 
interaction models like CANDE. In Chapter 5 it was shown that CANDE' 
predictions for out-of-ground box culverts loaded in bearing correlated 
very closely with the analytical predictions subsequently used to develop 
the ASTM design tables (14). Thus for buried boxes, it may be presumed 
that comparisons between CANDE and ASTM design tables will be influenced 
primarily by the modeling of soil and soil-structure interaction as op- 
posed to the modeling of the box. 

The comparisons reported herein are divided into two main sections; 
(1) dead loading due to soil weight only (ASTM C789, Table 3), and (2) 
dead loading due to soil weight plus HS20 live loading conditions (ASTM 
C789, Table 1). 

8.1 BOX SECTION STUDIES FOR DEAD LOAD 

Table 3 of ASTM C789 lists the design earth cover (allowable fill 
height) for each standard box size as a function of- the steel reinforce- 



85 



ment areas A ., ^c?* ^cV anc * ^qa' ^ or this study, a subset of these 
standard boxes were selected covering the typical range of box spans, 
rise/span ratios and amounts of steel reinforcement. These subsets 
are shown in Tables 8.1a and 8.1b. 

Table 8.1a represents the typical range of box spans; large (10 foot 
span) , intermediate (8 foot span) and small (4 foot span) where the 
span/rise ratio is an intermediate range 1.3 to 1.7. For each box, three 
levels of steel area (low, medium, and high) are listed and correspond 
to increased levels of design earth cover. In a similar manner, Table 
8.1b identifies three standard boxes with span/rise ratios ranging from 
1.0 to 2.0 and a common box span of 8 feet (2.44 m) . Taken together, 
Tables 8.1a and 8.1b cover the typical range of the standard ASTM box 
designs. Note that the intermediate box 8*6-8 (span*rise- inches wall 
thickness) is common to both tables. Thus, there is a total of 5 different 
box sizes with three levels of reinforcement, providing 15 different box 
sections for comparative analysis. 

Comparison Objectives and CANDE Model . For each of the ASTM box sections 
defined above, CANDE predictions are compared with ASTM assumptions for 
(a) soil load distribution on box at design earth cover, and (b) soil 
load distribution on box at failure cover heights. In addition, the 
consistency of ASTM designs are evaluated with CANDE with regard to 0.01 
inch cracking load and failure load. 

In order to make these comparisons, the parameters of the CANDE 
model are defined as consistantly as possible with ASTM assumptions. 
The concrete properties assumed for each box are: 

f£ = 5000 psi (345000 kPa) compressive strength 

e t = 0.0001 in/in cracking tensile strain 

Y c = 150 lbs/ft 3 (23.5 kN/m 3 ) weight density 

The remaining concrete parameters are taken as the standard values 
in Table 3.1. 



86 



TABLE 8.1 Reinforcement of Concrete Box Culverts Under Earth 

Dead Load Conditions (ASTM Table 3) Used for Comparison 

(a) Span/Rise Approximately 1.5, Span = large, intermediate and small 



BOX 


ASTM Design 
Earth Cover 
(ft) 


Reinf . 


si 
(in 2 /in) 


s2 
(in 2 /in) 


A Q 

s3 
(in 2 /in) 


A / 
s4 

(in2/in) 


















6 


Low 


.02000 


.02000 


.02000 


.02000 


10*6-10 


10 


Medium 


.02333 


.02833 


.03000 


.02000 




14 


High 


.03250 


.03833 


.04000 


.02000 


















6 


Low 


.01583 


.01583 


.01667 


.01583 


8*6-8 


10 


Medium 


.01667 


.02417 


.02583 


.01583 




14 


High 


.02333 


.03333 


.03500 


.01583 


















10 


Low 


.01000 


.01000 


.01083 


.01000 


4*3-5 


14 


Medium 


. 01000 


.01417 


.01000 


. 01000 




18 


High 


.01167 


.01833 


.01833 


.01000 



(b) Intermediate Span, Span/Rise from 1.0 to 2.0 



BOX 


ASTM Design 
Earth Cover 
(ft) 


Reinf. 


A i 
si 

(in 2 /in) 


A 
s2 

(in2/in) 


A O 

s3 
(in 2 /in) 


A / 
s4 

(in 2 /in) 


8*4-8 


6 
10 
14 


Low 
Medium 
High 


.01583 
.02000 
.02833 


.01583 
.02166 
.02917 


.01583 
.02250 
.03000 


.01583 
.01583 
.01583 


8*6-8 


6 

10 
14 


Low 

: Medium 
High 


.01583 
.01667 
.02333 


.01583 
.02417 
.03333 


.01667 
.02583 
.03500 


.01583 
.01583 
.01583 


8*8-8 


5 

8 

12 


Low 

Medium 
High 


.01583 
.01583 
.01750 


.01583 
.02167 
.03083 


.01667 
.02333 
.03333 


.01583 
.01583 
.01583 



1 ft = 0.3048 m 
1 in = 2.54 cm 



07 



and 



Assumed steel properties are: 

f y = 65000 psi (448000 kPa) yield stress 
E s = 29000 ksi (200000 MPa) Young's modulus 



T c = 1.25 in (3.18 cm) concrete cover to steel center 



S^ = 2.00 in (5.08 cm) longitudinal spacing for crack prediction 

An example of the CANDE input parameters for 8x6 box is given in 
Appendix C. 

Since the ASTM approach does not consider soil stiffness, the CANDE 

solutions use two soil conditions, soft and stiff, for the analysis of each 

box, thereby bracketing the practical range of soil stiffness. Soil moduli 

values for soft and stiff conditions are given in previous chapter in Table 

3 
7.2. For both conditions, soil density is taken as 120 pcf (18.8 kN/m ). 

All CANDE solutions are obtained using the new level 2 box generation 
scheme for an embankment installation. Nine construction increments of 
soil are used to-bring the soil height up to the ASTM design cover height 
to facilitate the comparison of loading distributions assumed by ASTM with 
those predicted by CANDE. Thereafter, additional soil layers are added 
until flexural failure is observed. During this loading sequence, the 
cover height causing initial 0.01 inch cracking is determined along with 
the cover height causing shear failure, providing shear failure occurs 
before flexural failure. 

Load Distribution Comparisons at Design Cover Height . The ASTM assumed 
load pattern due' to soil pressure and box weight are shown in Figure 8.1. 
Vertical soil pressures are assumed uniform and proportional to cover 
height. Lateral pressures are assumed to vary linearly, dependent on 
the coefficient of lateral earth pressure generally assumed to be 0.5. 
No shear traction on the box sides is assumed in the ASTM pattern. 

Figure 8.2 illustrates, the nature of a typical load distribution 
predicted by CANDE resulting from soil loading and box weight. Vertical 
soil pressures are not uniform, lower in the middle where bending defor- 
mation is greatest. Lateral pressure along the box increases with depth 



88 



Fs 
uj c 



{ :l t 



<Jc 
C"s 



t ■> ;' 



Z y ; t, :, i. j, y I i. Z — 



f—s 



r - r 



m 



2 ,i 4 ' + ** T 7 '< 



*-* 



H- 



Ghs 



R 



1 G"wn 



(7s = X* s x H 
Oc=W/(S+2t) 

aic = !fcx t 
(Tms = 0.5(Cs) 
(Thu^Ghs* 0.5 3s(R+t) 



W = Total Weight of 
Box 

S =• Span of Box 



R = Rise of Box 

H = Height of Soil 
from. Top of Box 

Oc = Unit Weight of 
Concrete 

#S = IJnit Wei Sat of 
Soil 

+ s Thickness of Box 

0< = Coefficient for 
Lateral Pressure 



Figure 8.1 Loading of a Box Culvert Due to Soil According 
to ASTM Norms. 




UJvy= 



OJH UJ w = 



Sv= 



Normal Load in the 
Vertical Direction 

Normal Load in the 
Horizontal Direction 

Shear Load Over the 
Walls 



Figure 8.2 Typical Load Pattern Due co Soil Load Obtained from CANDE. 



89 



but not linearly. This applies to both sides of the box but is only 
illustrated on the right side in Figure 8.2. In addition to the normal 
pressures, significant shear traction develops over the side walls 
acting mostly downward. This is illustrated on the left side of the box. 
Shear traction on the top and bottom slabs is also present but is not 
significant and is not shown. Shear traction on the side walls can 
amount to 50% of the net downward force which must be equilibrated by 
the pressure along the bottom slab. This is an effect not considered 
in the ASTM load pattern and should be kept in mind in the subsequent 
comparisons. 

In order to compare ASTM and CANDE load distributions at design 
earth cover, normalized plots are constructed by dividing the CANDE 
predictions by the ASTM assumption at each point around the box. This 
is clarified in Figure 8.3 where the dashed lines represent normalized 
values of unity, and the solid lines represent the ratio of CANDE pre- 
diction to ASTM assumption. Shear traction is arbitrarily normalized 
by dividing the CANDE prediction for shear traction by the ASTM assump- 
tion for normal pressure on the top slab. Due to symmetry, both sides 
on the box experience identical loading distributions. Normalized 
plots for lateral soil pressure are shown on the right side of the box, 
while normalized plots for shear traction are shown on the left. 

With the above understanding, Figures 8.4 through 8.8 show the 
normalized load distributions for each box in Table 8.1a,b. Each 
figure shows six normalized plots per box representing the three levels 
of reinforcement and the two soil conditions. 

In general, CANDE predictions for the normal pressure on the top 
and bottom slabs are not uniform, increasing from the center of the 
slab to the corner of the box. Normal pressure at the center of the 
top slab are very close to ASTM assumption and increases to a range of 
20% to 30% greater than the ASTM assumption near the corner, depending 
on the soil conditions and level of reinforcement. CANDE predictions for 
the normal pressure on the bottom slab is significantly higher for soft 
soil than stiff soil. This is because soft soil generates greater shear 



90 




^ 



Olw = 



V= ^v/ffs+UJc) 



Figure 8.3 - Explanation of the Plots for Normalized Normal 
Pressure and Shear Forces Acting on the 3ox 
(CANDE Prediction/ ASTM Assumption) . 



91 




(a) Low reinforced and 
stiff soil 



UJ V 





UJ V 

(b) Low reinforced and 
soft soil 




(c) Medium reinforced and 
stiff soil 



UJ V 



(d) Medium reinforced and 
soft soil 



UJ V 




Hs 14.0 f" 



(e) High reinforced and 
stiff soil 



vu, 



UJy 




(f) High reinforced and 
soft soil 



Figure 8.4 - Normalized Plots for Normal Pressure and Shear Acting 
on a 10x6 Box iZ ASTM Height of Soil 



92 




(a) Low reinforced and 
stiff soil 




(c) Medium reinforced and 
stiff soil 




(e) High reinforced and 
stiff soil 



Ulu 




I UJ, 



(b) Low reinforced and 
soft soi 




-PH 



(d) Medium reinforced and 
soft soil 




(f) High reinforced and 
soft soil 



Figure 8.5 - Normalized Plots for Normal Pressure and Shear Acting 
ou a 8x6 Box ac ASTM Height of Soil. 



93 




UJH 




(a) Low reinforced and 
stiff soil 



UJ V 

(b) Low reinforced and 
soft soil 



UJv 




UJ V 



(c) Medium reinforced and 
stiff soil 




(d) Medium reinforced and 
soft soil 





UJ V 



(f) High reinforced and 
soft soil 



(e) High reinforced and 
stiff soil 

Figure 8.6 - Normalized Plots for Normal Pressure and Shear Acting 
on a 4x3 Box at ASTM Height of Soil. 



94 




(a) Low reinforced and 
stiff soil 



UJ H 








I 

i 


H=6.0 ft 


Ivjl 


\ 


{ 








1 


-"v "** 





(d) Lov reinforced and 
soft soil 



UJ V 




(c) Medium reinforced and 
stiff soil 




UJh 



(d) Medium reinforced and 
soft soil 



UJ V 




UJy_ 



(e) High reinforced and 
stiff soil 



ujh 




oj v 

(f) High reinforced and 
soft soil 



Figure 8.7 - Normalized Plots for Normal Pressure and Shear Acting 
on a 8x4 Box at ASTM Height of Soil. 



| UJh 



95 





(a) Low reinforced and 
stiff soil 



(b) Low reinforced and 
soft soil 



uj v 




UJv 



(c) Medium reinforced and 
stiff soil 




(d) Medium reinforced and 
soft soil 



UJ V 




UJv 



(e) High reinforced and 
stiff soil 




UJv 

(f) High reinforced and 
soft soil 



Figure 8.8 - Normalized Plocs for Normal Pressure and Shear Acting 
on a 8x8 Box at ASTM Height of Soil. 



96 



forces over the side walls producing a greater downward force. For the 
stiff soil condition CANDE predictions are similar to ASTM assumption at 
the center of the bottom slab and increases to about 60% to 70% greater 
than the ASTM assumption near the corner. For the soft soil condition 
CANDE predictions are about 20% to 40% greater than the ASTM assumption 
at the center of the bottom slab and increases to about 70% to 100% 
greater near the corners. The lateral pressure from CANDE predictions 
are not linear like the ASTM assumption, but in general the magnitudes 
are close. 

Load Distribution Comparisons at Failure Cover Heights . To further this 
study, each box was loaded beyond the ASTM design earth cover to failure 
for both soil conditions. Figures 8.9 to 8.13 show load-deflection histories 
of all the boxes analyzed to failure. For each box shown, the type of 
failure that first occured is indicated at the height of soil cover where 
failure occured. As expected, the boxes buried under soft soil conditions 
exhibit greater deflections during their load history and fail prior to 
the identical box analyzed in stiff soil conditions. 

For fill heights at failure, normalized load-distribution plots 
(CANDE prediction divided by the ASTM assumption) are constructed in 
the same manner as previously described and are shown in Figures 8.14 to 
8.18. Note, the magnitude of the ASTM load distributions are linearly 
related to cover height but retain the same shape for all fill heights. 
Load distributions from CANDE change both in magnitude and in shape 
during loading as a consequence of soil-structure interaction and 
changing stiffness of the box. 

The normalized plots at failure show the same general trends as the 
normalized plots at design earth cover. Now, however, the normal pressure 
distributions on the top and bottom slab tend to increase more rapidly, 
beginning with relatively smaller magnitudes at the slab centers and in- 
creasing to relatively higher magnitudes at the slab corners. This is 
attributed to the reduction of slab bending stiffness as failure develops, 
i.e., a greater portion of the soil load is shifted to the stiff er corners 
where the side walls serve as thrust columns. 



97 



.0041 

.008 
.012 

.016 

.020 
.024 
& 



soft SO i 



'/r 




o 

+ 



high reinf 
medium reinf 
low reinf 
bonding failure 
shear failure 



(a) Vertical deflection/rise vs. fill height /rise. 



.002 

.0041 

.006 

.008-1 
Sh 



1 2 3 

1 i ~ 



5 6 H /r 




iff soil 



SOft SOil 



'/s 

(b) Horizontal deflection/ span vs. fill height/rise. 



Figure 8.9 - 



Height of Soil - Deflection for 10x6 Box with 
Soil Loads. 



98 



3 4 5 H/ R 



.002-1 

,004 
.006 
.008 
.010- 

.012- 
H 



Stiff SCi 



SOft SOi 




high re inf. 

medium reinf 

low reinf 

o bending failure 
. + shear failure 



(a) Vertical deflection /rise vs. fill height /rise. 



5 HA 



.002 

.oo4^ 

.006 



Ws 




stiff so 



soft so 



(b) Horizontal deflection/ span vs. fill height/rise. 



Figure 8.10 - Height of Soil - Deflection for 3x6 3ox with 
Soil Loads. 



99 



8 10 H /R 



SOft SOil 




S^^stif f SOii 

\ \ 
\ \ 
\ \ 


high reinf 




I 


medium reinf 




1 


low reinf 




i 


bonding failure 






+► shoar failure 







.002- 

.004- 
.00 G 
.008- 
.010- 
.012 
Sv /R 

(a) Vertical deflection/ rise vs. fill height/rise- 



.001 

.002 

£03-1 



ft 



4 /s 



Stiff SOil 




10 H/ R 



soft so 



(b) Horizontal deflection/ span vs.. fill height /rise. 

Figure 8.11 - Height of Soil - Deflection for 4x3 Box with 
Soil Loads. 



100 




10 h /r 



.030- 



high rsinf \ 

m odium rsinf j 

low roinf? \ 

o bonding failursj 

+ shoar failure 



9 



(a) Vertical deflection/rise vs. fill height/rise. 



.002- 
,004- 
.006- 



r. 



% 



Stiff SOi 




10 H/ 



R 



soft so 



(h) Horizontal deflection/ spaa vs.: fill height/rise. 



Figure 8.12 - Height of Soil - Deflection for 8x4 Box with 
Soil Loads. 



101 



2 3 



h /p 



.002 
.004 
.006 
.008 
.010 
.012 



■ , ~^- g ^' ' L " 

SOft SOil^ \\ \ 

V 


\ 
\ 

"\ 
I 
i 1 


i ' ii 
^rStif f SOil 


high reinf 






medium reinf. 






low reinf. 






o bonding failure 






+• shear failure 







(a) Vertical deflection/rise vs. fill height /rise, 



1 2 3 4 5 H / R 



soft soi 




stiff soil 



.001 

.002- 

.003- 

fys 

(h) Horizontal deflection/span vs. fill height /rise. 



Figure 8.13 - Height of Soil - Deflection for 8x8" Box with 
Soil Loads. 



102 




UJv 



UJ V 



(a) Low reinforced and 
stiff soil 




ULly 



(b) Low reinforced and 
soft soil 





(c) Medium reinforced and 
stiff soil 



UJv/ 

(d) Medium reinforced and 
soft soil 




(e) High reinforced and 
stiff soil 




(f ) High reinforced and 
soft soil 



Figure 8.14 - Normalized Plots for Normal Pressure and Shear Acting 
on a 10x6 Box at Failure Load. 



103 




(a) Low reinforced and 
stiff soil 




(c) Medium reinforced and 

stiff soil 




(b) Low reinforced and 
soft soil 



UJy 




(d) Medium reinforced and 
soft soil 



Note: The 8x6 box with high reinforcement has plots similar to the medium 
reinforced, where the failure is due to shear. 



Figure 8.15 - Normalized Plots for Normal Pressure and Shear Acting 
on a 8x6 Box at Failure Load. 



104 




(a) Low reinforced and 
stiff soil 




(c) Medium reinforced and 
stiff soil 




(b) Low reinforced and 
soft soil 




(d) Medium reinforced and 
soft soil 



Note: The 4x3 box with high reinforcement has plots similar Co the 
medium reinforced, where the failure is due to shear. 



Figure 8.16 - Normalized Plots for Normal Pressure and Shear Acting 
on a 4x3 Box at Failure Load. 



105 



UJ V 



uu v 









rr 




i 






\ 


|\ 


H=22.0 ft . 


1 
1 


jS v 


1 


1 
1 


3/^ 




(a) Low reinforced and 


stlf 


f soil 





UJh- 




uj h 



(b) Low reinforced and 
soft soil 





(c) Medium reinforced and 
stiff soil 



UJ V 



(d) Medium reinforced, and 
soft soil 



Noter The 8x8 box with high reinforcement has plots similar to the 
medium reinforced, where the failure is due to shear. 



Figure 8.17 -Normalized Plots for Normal Pressure and Shear Acting 
on a 8x8 Box at Failure Load. 



106 





(a) Low reinforced and 
stiff soil 



(b) Low reinforced and 
soft soil 




(c) Medium reinforced and 
stiff soil 




(d) Most reinforced and 
soft soil 




(e) High reinforced and 
stiff soil 




Ol v 

(f ) High reinforced and 
soft soil 



Figure 8.18 - Normalized Plots for Normal Pressure and Shear Acting 
on a 8x4 Box at Failure Load. 



107 



Lateral pressure on the side wall tend to be greater than ASTM 
predictions, particularly in the center region where outward deflections 
mobilize passive soil resistance. Side wall shear traction is maximum 
at the top walls acting in the downward direction. Near the bottom, 
shear traction reverses sign, but the net effect is a significant down- 
ward force, an effect not considered in the ASTM assumed load pattern. 

Cracking and Failure Loads Comparisons . The CANDE prediction for the 
fill height producing 0.01 inch cracking is shown in Tables 8.2 and 8.3 
(second column from end) corresponding to the boxes defined in Tables 8.1a 
and 8.1b for both soft and stiff soil conditions. For each box size it 
is observed that the cracking load (fill height) increases with the 
level of reinforcement. Also for identical box cross sections, the 
cracking load increases with soil stiffness. 

To check if the ASTM design earth covers are conservative compared 
to predicted fill heights at which 0.01 inch cracking occurs, a fill- 
height ratio (CANDE prediction/ASTM design cover) is shown in the last 
columns. This ratio should be more than 1.0 for conservative designs. 
For stiff soils, the cracking load ratio varies from 1.06 to 2.02 im- 
plying the ASTM design covers are conservative. For soft soils, the 
ratio varies from 0.75 to 1.44 implying some designs may not be conser- 
vative. Boxes with low reinforcement tend to be more conservative than 
with high reinforcement. As a general conclusion, the ASTM boxes are 
moderately conservative with respect to 0.01 inch cracking at design 
earth cover providing good quality soil is used. 

Also shown in Tables 8.2 and 8.3 are the CANDE predictions for 
fill height at failure as controlled by flexure or shear. In most 
cases shear failure occurs prior to flexure failure except for some 
lightly reinforced boxes. Identical boxes fail at lower fill heights 
in soft soil than in stiff soil. Both shear and flexure failure heights 
are reduced in soft soils, but flexure failure heights are reduced by 
a greater precentage. 

A "failure load ratio" is defined here by dividing the predicted 
failure height (as controlled by shear or flexure) by the ASTM design 



108 



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110 



earth cover and is tabulated in the center column of the tables. Pre- 
sumably, the ASTM designs are based on a load factor of 1.5 times the 
design earth load. Thus, the failure load ratio defined above should be 
at least 1.5 to achieve the intended ultimate capacity. For stiff soils, 
the ratio varies from 1.5 to 4.4, whereas for soft soils, the ratio 
varies from 1.3 to 3.6. For a given soil condition and box size, the 
ratios are higher for low reinforcement than for high reinforcement. 
Overall, it is concluded that the ASTM box designs are conservative with 
respect to the 1.5 load factor criterion when good quality soil is used, 
but less so for the high reinforcement than low reinforcement. In other 
words, the ASTM design earth cover specified for a box with low reinforce- 
ment is more conservative than the specified earth cover for the identical 
box with high reinforcement. 

8.2 BOX SECTION STUDIES WITH LIVE LOADS 

In this section the effect of live loads on shallowly buried boxes 
are investigated and compared with ASTM C789 design tables. 

The ASTM Specifications consider two types of live load in their 
box culvert design tables, HS-20 truck loads (ASTM Table 1) and inter- 
state truck loads (ASTM Table 2). Due to the small difference between 
these design tables, only the HS-20 live loads are considered in this 
study. Figure 8.19 shows the HS-20 truck axle loads along with an 
"equivalent" transverse strip load P = 222 lb/in (389 N/cm) used as a 
reference plane strain loading in the CANDE analysis. The strip load P 
represents the static weight of the middle axle tire loads distributed 
by the axle length as shown in the Figure. 

The box-soil system analyzed in CANDE is an embankment installation 
using the Level 2 box automatic mesh generation along with the extended 
level 2 option for defining the live loads. As before, two types of soil, 
stiff and soft (see Table 7.2), are used for each box culvert. Figure 8.20 
shows a typical box culvert cross section, where the live load P repre- 
senting the HS-20 truck's middle axle is applied over tne center of the 
box culvert. The other axles are well away from the box culverts con- 



Ill 




B 



8000 lb 320001b 320001b 



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HS-20 Live Load 



f-^-4 



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5 ISOOOIb 160001b 
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144 in 



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Figure 8.19 - HS-20 Truck Live Load and Equivalent Plane Strain 
Strip Load. 



P=222 1 b/ in 








(/ns/t- 



4- 




Figure a. 20 - Typical Box Culvert Cross Section Used for Live Load 
Comparison Study. 



112 



sidered herein and have negligible influence on the box deformations. 

To study the effect of live load at shallow soil cover, a set of 
five boxes representing a range of sizes and span/rise ratios were 
selected from ASTM Table 1. Each of these boxes has as a minimum 
allowable fill height of 2.0 feet (0.61 m) to disperse the concentrated 
live load. Table 8.4 lists these boxes and the ASTM steel reinforcement 
areas specified for the common minimum soil cover. For the CANDE analysis 
each box was incrementally loaded with soil layers up to 2.0 feet 
cover height. Next, the live load P was applied and then incrementally 
increased to a value P* at which a 0.01 inch crack occurred in the box 
culvert. This was repeated for each of the five boxes using stiff and 
soft soil conditions. 

By forming the load ratio P*/P, the ASTM designs can be evaluated 
on the basis of 0.01 inch cracking design criterion. Table 8.5 shows 
the results of this study. Note that the load ratios are high but 
fairly uniform, ranging from 4.0 to 4.7 including both soil conditions. 
Thus, it is concluded that ASTM designs at 2.0 ft cover heights tend to 
be overly conservative even if impact loads are added to the HS-20 
loading (ASTM recommends impact loads up to 20%) . 

Studies similar to the above indicated that the influence of live 
loads is negligible compared to dead loads for fill heights greater 
than eight feet. 



113 



Table 8.4 Reinforcement of Box Culverts with Minimum 
Soil Cover and HS-20 Live Load (ASTM Table 1) 



BOX 


ASTM 

H soil 
(ft) 


. A si 

(in 2 /in) 


A S2 
(in 2 /in) 


A S3 
(in 2 /in) 


A S4 
(in2/in) 














4*3-5 


2.0 


.01750 


.02250 


.02000 


.01000 


8*6-8 


2.0 


.02583 


.03833 


.02917 


.01583 


10*6-10 


2.0 


.02917 


.03833 


.02833 


.02000 


8*8-8 


2.0 


.02167 


.04250 


.03333 


.01583 


8*4-8 


2.0 


.Q3Q83 


.03333 


.02417 


.01583 



Table 8.5 Live Load Performance Factor for Box Culverts 
with Minimal Soil Cover and HS-20 Live Load 



BOX 


ASTM 

H soil 
(ft) 


HS-20 

P 
(lb/ in) 


Type of 
Soil 


* 

P 

(lb/in) 


P*/P 














4*3-5 


2.0 


222 


STIFF 
SOFT 


1046 
942 


4.71 
4.24 


'8*6-8 


2.0 


222 


STIFF 
SOFT 


1006 
942 


4.51 
4.24 


10*6-10 


2.0 


222 


STIFF 
SOFT 


1030 
930 


4.64 
4.19 


8*4-8 


2.0 


222 


STIFF 
SOFT 


1022 
892 


4.60 
4.02 


8*8-8 


2.0 


222 


STIFF 
SOFT 


926 
880 


4.17 
3.96 



*Result obtained from CANDE 



1 ft = 0.3048 m 
1 lb/in = 1.22 N/m 



114 



CHAPTER 9 

SOIL MODELS 

Soil models originally incorporated into the CANDE-1976 program 
included: (a) linear elastic (isotropic or orthotropic) ; (b) incremental 
elastic, wherein elastic moduli are dependent on current fill height 
(overburden dependent); and (c) variable modulus model using a modified 
version of the Hardin soil model. The latter model employs a variable 
shear modulus and Poisson's ratio which are dependent on maximum shear 
strain and hydrostatic pressure (1). 

The purpose of this chapter is to discuss the implementation of a 
new soil model into the CANDE program, called here the Duncan soil model, 
and to present standard parameters for characterizing this model. The 
Duncan soil model has had a substantial history of development and appli- 
cation over the last decade (26, 27, 28, 29 and associated references). 
It is a variable modulus model such that increments of stress are re- 
lated to increments of strain by the isotropic form of Hooke's law wherein 
the elastic paremeters are dependent on the stress state. For plane 
strain, this incremental relationship may be written as: 



Aa 

x 

Aa 

y 

At 



11 
12 



C 12 ° 




Ae x 


c 22 




Ae y 


o c„ 




ay 



9.1 



where Aa .,Aa ■ normal stress increments 

x y 

At = shear stress increments 

Ae .,Ae - = normal strain increments 
x y 

Ay ■ shear strain increments 
C ■ constitutive matrix components (variable) 

In accordance with Hooke's law (isotropic form), the matrix com- 
ponents C . . are all defined with any two elastic parameters. Table 9.1 



115 



Table 9.1 Elastic Equivalents for 
Isotropic Plane Strain 



Matrix 
Component 


(E, v) 


(E, B) 


C ll - C 22 


E(l-v) 


3B(3B + E) 
9B - E 


(1+v) (l-2v) 


C 12 


Ev 


3B(3B - E) 
9B - E 


(1+v) (l-2v) 


C 33 


E 


3BE 
93-E 


2 (1+v) 



116 



shows this relationship for the elastic parameters pertinent to this 
study: Young's modulus and Poisson's ratio (E, v) and Young's modulus 
and bulk modulus (E, B) . If (E, v) or (E, B) are described as a function 
stress, characterizing the nonlinear behavior of soil, then the matrix 
components C.. are also defined and infer tangent relationships between 
stress and strain increments. 

9.1 DUNCAN MODEL REPRESENTATION OF ELASTIC PARAMETERS 

Initially, Duncan and his colleagues characterized soil behavior 
with a variable tangent Young's modulus E t and a constant Poisson's 
ratio where Fj- employed the so-called hyperbolic stress-strain model (27) 
Subsequently, a variable Poisson's ratio formulation was introduced to 
better represent the volume change behavior observed in triaxial soil 
tests (26,28). Recently, a tangent bulk modulus formulation was intro- 
duced to replace the variable Poisson's ratio (29). 

The last model, which employs tangent Young's modulus and tangent 
bulk modulus formulations, is adopted for this study and incorporation 
into CANDE. An extensive evaluation of the variable Poisson ratio 
formulation was undertaken during the course of this study and was 
found to behave erratically in some cases (30). Consequently, it was 
not incorporated into the CANDE program. 

Development details of the Duncan model are well documented else- 
where (29). The final expressions for tangent Young's modulus and 
bulk modulus as a function maximum and minimum principle stresses for 
loading conditions are given here. 

'The tangent Young's modulus expression is: 

a R (l-sin<f>)(a -a ) 

E = KP (^) [1 - — — - X . J 9.2 

t a P 2 c cos<}> + 2a sxn<{> 

3. -J 

where a-j = minimum principle stress (compression positive), 

o"! = maximum principle stress (compression positive). 

P a = atmospheric pressure (for dimensionless convenience) . 

K = modulus number, nondimensional 



117 




n = modulus exponent, typical range -1.0 to 1.0. 

Rf = failure ratio, typical range 0.5 to 0.9. 

c = cohesion intercept, units same as P a . 

<j> = friction angle, radians 

A<J> = reduction in <J> for 10-fold increase in 03. 

(i.e., 4 - <j) -A(J» log ,_A 

xu ^Pa' 

The tangent bulk modulus expression is a function of minimum compressive 
stress given by: 

9.3 

where K_ = bulk modulus number, dimensionless. 

m = bulk modulus exponent, typical range 0.0 to 1.0. 

In Equations 9.2 to 9.3, there are a total of eight parameters to 
define a particular soil in loading: K, n, Rf, c, <£ , and A<J) to define 
E t ; and K^ and m to define B t . Established methods for determining 
these parameters from conventional triaxial tests have been reported by 
Duncan and his colleagues (28,29). In the last section of this chapter, 
conservative estimates of these parameters are given for various soil 
types and degree of compaction. 

The behavioral characteristics and limitations of the Duncan soil 
model (Equations 9.2 and 9.3) are enumerated below along with the 
programming strategy used in the Duncan finite element program called 
SSTIPN. 

(1) As 03 increases (e.g. confining pressure in a triaxial test) 

E t and B t becomes stiffer (assuming m and n are greater than 

zero). However, as maximum shear stress increases (i.e. 

(a. - cO/2), E t becomes weaker, but B«- remains constant. Such 
1 £ u 

behavior is typical of triaxial tests on which the model was 
developed. 

(2) Shear failure is said to occur when E t approaches zero. That 
is, the bracketed term in Equation 9.2 approaches zero as 



118 



a - a increases. If a significant portion of the soil mass 
fails in shear, the results may no longer be reliable because 
the model is not applicable for soil instability. To avoid 
numerical problems, the SSTIPN algorithm arbitrarily limits 
the minimum value of the bracketed term in Equation 9.2 to 
1 - .95 Rf. Thus, Et does not actually become zero in shear 
failure. 

(3) Tension failure is said to occur when a 3 becomes tensile. In 
such cases the soil stiffness breaks down and cannot carry load. 
To cope with this problem, the SSTIPN algorithm computes a small 
value for bulk modulus from Equation 9.3 by specifying a /P = 0.1 

3 Si 

whenever 03 is tensile. For the second elastic parameter, 
Poisson's ratio is arbitrarily assigned the value 0.495 and 
Equation 9.2 is ignored. This results in equivalent Young's 
modulus whose value is approximately 3% of the bulk modulus. 

(4) In addition to the special treatment for shear and tension 
failures, the SSTIPN algorithm sets limits on B t as predicted 
from Equation 9.3 dependent on the value of E t from Equation 9.2. 
Specifically, B t = E t /3.0, if B t is less than this value, and, 

Bt = 34.0 E t , if B t is greater than this value. These limits 
correspond to maintaining the equivalent Poisson's ratio within 
the range 0.0 to 0.495. 

(5) For each load step (e.g. construction increment), the SSTIPN 
algorithm utilizes two iterations to determine B^ and E t as 
defined above. For the first iteration, the stresses existing 
in the element at the end of the previous load step are used 
to estimate B t and E t to obtain approximate stress increments. 
The second iteration repeats this solution wherein B t and E t 
are now determined by adding one-half of the stress increments 
determined in the first iteration to the previous stress state. 
Upon completion of the second iteration, the stresses are accumu- 
lated and printed out, and the next load step is considered. No 
convergence check is made. 



119 



(6) When an element first enters the system, the "existing" stress 
state used to determine B t and E fc for the first iteration is 
determined in a special manner depending on whether the element 
is part of the initial system (e.g. pre-existing foundation) or 
part of a new construction increment. For the case of elements 
belonging to the initial system, existing stresses are defined 
by the user (input), or if the foundation is composed of hori- 
zontal rows, initial stresses can be automatically approximated 
by overburden pressure and a lateral coefficient. 

Elements belonging to a new construction increment do not 
have an existing stress state prior to entering the system. 
However in order to evaluate E t and B t for the first iteration, 
the SSTIPN algorithm estimates initial stresses based on element 
height, soil density, assumed Poisson's ratio, and humped surface 
angle. 

(7) According to published reports (28,29), "unloading" of the 

Duncan soil model is accomplished by replacing the tangent 

Young's modulus function (Equation 9.2) with an unloading 

expression; E u = K P a , 3vn, where 1^ is an unloading modulus 

( p ) 

number whose value is greater than K. 

Although this is relatively easy to program, there are 
serious theoretical objections to this description of unloading. 
Presumably, the criterion for unloading (i.e. switching from 
E t to E u ) is by observing a decrease in maximum shear stress 
irrespective of 03. Such a criterion may be sufficient for 
load paths where 03 is constant (e.g. triaxial test), however 
for more general load paths, serious violations of the contin- 
uity principle can occur, i.e., two arbitrarily close load 
paths should not result in dramatically different stress-strain 
responses. 

For this reason, the unloading function is not incorporated 
into the CANDE program. Further research on unloading is war- 
ranted. 



120 



9.2 CANDE SOLUTION STRATEGY FOR DUNCAN MODEL 

The CANDE algorithm for the Duncan soil model is contained in a 
new subroutine called DUNCAN. Here the representation of E t and B t 
(Equations 9.2 and 9.3), shear failure, and tension failure are treated 
in a similar fashion to the SSTIPN algorithm discussed in the previous 
section. However, there are some significant differences in the CANDE 
solution strategy with regard to (a) number of iterations, (b) averaging 
Et and B t over a load step, and (c) treatment of elements entering the 
system for the first time. These differences are discussed below. 

Iterations . As previously explained, the SSTIPN algorithm uses two 
iterations per load step for all loading schedules. Preliminary studies 
during this research indicated that using just two iterations can lead 
to serious error in predicting E and B even when load increments are 
relatively small (e.g. one layer of elements per construction increment). 

To deal with this problem, the CANDE algorithm allows the maximum 
number of iterations to be specified by the user. During the iteration 
process, the current estimate of Et for each element is compared percen- 
tage-wise with the previous estimate of E t . *If two succeeding estimates 
of E t converge within a specified error tolerance for all elements, the 
iteration process is terminated and algorithm advances to the next load 
step. Should convergence not be achieved after the specified maximum 
number of iterations, a warning message is printed out prior to advancing 
to the next load step. 

Note, the convergence check is only considered for E t , not B t . 
However, it may be presumed that B t converges more rapidly than Et 
since the former is only a function 03, whereas the latter is a more 
sensitive function dependent on a-~ and 03. 

Averaging Et and B t . Equations 9.2 and 9.3 are tangent moduli expressions 
for E t and B t for a particular principle stress state a-^ and 03. As a 
load increment is applied, the stress state changes, inferring changes in 
Et and Bt. In order to adequately represent the effects of these changes 
in Equation 9.1, E and B should represent "average" values over the 
load step. This, of course, is the purpose of iteration. 



121 



One way of obtaining average values is to evaluate Et and B t based 
on the average stress state during the load step as is done in the SSTIPN 
algorithm. Alternatively, one may average E t at the beginning of the 
load step with E^- at the end of the load step. Likewise for B t . Speci- 
fically, this may be written as: 

E avg " a-r^'+rtj 9.4 

B avg " ( 1 - r > B l H ' rB 2 9 - 5 

where E , B = E , B at start-of-load-step (known) 

E ? , B = E , B at end-of-load-step (iteratively determined) 

r = averaging ratio, (generally r = 1/2) 

For reasons to be subsequently discussed, the CANDE algorithm employs 
the averaging scheme given by Equations 9.4 and 9.5. Comparison studies 
between the stress averaging scheme and the moduli averaging scheme were 
found to give nearly identical results for r = 1/2. 

The averaging ratio r is treated as a material input parameter in 
the CANDE program. Generally r = 1/2, however for pre-existing soil 
zones, r = 1 permits proper calculation of pre-existing stresses as dis- 
cussed next. 

Entering Elements . Soil elements enter the structural system in one of 
two categories. The first category apply s to pre-existing or in-situ 
soil elements in which an initial stress state exists but is unknown. 
Elements entering in this category are part of the initial configuration 
and belong to the first construction increment. 

The second category applys to fill soil elements, i.e., soil layers 
added to the system in a predefined construction schedule. Here, the 
initial stress state is non-existant prior to entry into the system. 
Both categories present special starting problems for the iteration 
procedures because the initial stress state is unknown or undefined. 

If pre-existing soil zones are to be characterized by the Duncan 
soil model, the 'initial stress state can be determined iteratively by 



122 



assuming the pre-existing soil zone is a construction increment loaded 
with its own body weight (and, if desired, a consolidation pressure). 
Here the averaging ratio should be set to 1.0, so that, E a and B a 
are equal to the end-of-load-step values E£ and B2, respectively, and 
correspond to the existing stress state. Beginning-of-load-step values 
E-l and B^ are initially set to 0.0 when an element enters the system. 
However when r = 1, they have no influence on the averaging process. 
After the first construction increment is complete, the program auto- 
matically sets the value of r to 1/2, so that, all subsequent moduli 
calculations represent load step averages. 

Elements entering the system in the second category have no initial 
stiffness prior to loading so that Ej_ - B]_ = 0. Accordingly, using 
r = 1/2 gives average moduli values equal to one-half of the end-of-load- 
step values, E and B . 

To start the iteration process for entering elements of either 
category, some guess must be made for E2 and B2 in order to construct 
the first trial stiffness matrix. This is achieved by arbitrarily de- 
fining "dummy" principle stresses from which initial estimates of E2 
and B2 are calculated. The dummy principle stresses have no effect on 
the final values of E2 and B2» however they do influence the number of 
iterations for convergence. Once an element has entered the system, the 
initial guess for E2 arid B2 for all subsequent load steps are equated 
to the last calculated values, thus dummy stresses are not required. 

Cande Algorithm . Figure 9.1 is a flow chart of the CANDE algorithm 
illustrating the solution strategy previously described. Some of the 
limit bounds are defined differently than in the SSTIPN algorithm. For 
example, the maximum equivalent Poisson ratio, v max , is set at 0.48 
rather than 0.495 in order to avoid unreasonably high values of C in 
Equation 9.1. Also, the shear failure factor (1-D) is assigned a lower 
limit of 0.05, rather than 1-0.95 Rf in order to provide a greater re- 
duction of stiffness in shear failure. 



123 



Incoming information, 
a , a - principle stresses 
i = iteration no. 




i 


■'/ 






Existing element 




New element 
E = 

B i = ° 

a, = 0.2 P 
1 a 


E 1 = E 2 

B 1 = B 2 
r = 0.5 


i = 1 














i > 1 

> 








a 3 = 0.1 P a 


.._ „. } 










1 


Check tension failure. 
a 3 < 


t w I 


yes 




i 




no 

< 


* 


) 


Set lower limit on a . 




Set tension values 




a > 0.1 P 
3 - a 




E 2 = (0.05) 2 K(.l) n 
B 2 = 1.67E 




* 


f 




Set limit on shear failure. 






D = R f ( a 1 " a 3 ) sin< J ) 






2 (c cos<j) + a sin<j>) 






< D <_ 0.95 






' 


t 




Compute end-of'-step moduli. 




E, = K(a./P ) n (l-D) 2 
i j a. 




B 2 = VW 






E 2 /3 £ B 2 < 8E 2 


' 








f 




<k 






1 


* 




Average moduli values. 




E = (l-r)E. + rE. 
avg 1 2 




B = (l-r)B n + rB„ 
avg 1 2 




.... . _ .. 1 


* 


fl ^-y Re P eat iteration. 


Check convergence of E„. 


E 2 (i) = E 2 (i+1) ? 


"^-si* 


c 


A Go 


to next loac 


[ step. 





Figure 9.1 CANDE Algorithm for Duncan Soil Model. 

124 



Two additional features of the CANDE algorithm not shown in Figure 9.1 
are; (1) an under relaxation scheme to improve the rate of convergence 
for E2, and (2) a constant Poisson ratio option which replaces the tangent 
bulk modulus formulation. 

The under relaxation scheme comes into play after the second iteration 
wherein each estimate of E£ is a weighted average of the current estimate 
and the previous estimate. This feature takes advantage of the observation 
that E2 generally converges in an oscillatory manner. 

When the constant Poisson ratio option is exercised, all references 
to the bulk modulus formulation are bypassed. Otherwise, the algorithm 
is essentially the same. 

9.3 STANDARD HYPERBOLIC PARAMETERS 

Whenever possible, the hyperbolic parameters characterizing the 
Duncan soil model should be determined directly from triaxial tests 
using established curve-fitting procedures (28,29). In many instances, 
however, triaxial data may be unavailable, and so, it is convenient to 

establish "standard" parameter values for various types of soil and 
degrees of compaction. Table 9.2 (abstracted from Reference 29) provides 
parameter values for four soil classifications, each with three levels 
of compaction. These "standard" values are conservative in the sense 
that they are typical of lower values of strength and moduli observed 
from numerous triaxial tests for each soil type. An independent study (30) 
to establish standard parameters for E t utilizing the same data base is 
in good agreement with Table 9.2. 

For convenience, the hyperbolic parameters in Table 9.2 are stored 
in CANDE and may be used by simply identifying soil type and level of 
compaction. 

The behavior of the Duncan soil model for simulated uniaxial strain 
and triaxial loading tests is shown in Figures 9.2 through 9.5 for the 
standard parameters in Table 9.2. Specifically, Figure 9.2a shows axial 
stress vs. axial strain in confined compression for three compaction 
levels of coarse aggregates. The slope of these curves is the tangent 



125 



CO 

5-i 
CD 
4-1 
CD 
6 
CO 
5-i 
CO 
Pi 



O 
X> 
5-i 

0) 

a 

T3 
5-1 

CO 

§ 

4-1 

CO 



CM 

w 



B 


CN CN CM 
... 

o o o 


o o o 

• • • 

o o o 


m m m 

dod 


CM CM CM 

• • • 

o o o 


*f° 


m m o 
r*» r^ m 
H 


o o o 
m m m 
<r cm iH 


o m o 
o r>» m 

CM 


o o o 
<t co m 

rH 


14-1 

Pi 


r — r~^ r^ 

CD CD o 


r — r^* r— ~ 

d o d 


r>. f*. r» 

• • • 

o o o 


r» r>« r-. 
odd 


c 


sr -a- -d- 

• • • 

o o o 


m m m 

CN CM CM 

• • • 

o o o 


vO vO vO 

... 

o o o 


m m m 
-si- <■ -a- 

odd 


w 


o o o 
o o o 

VO CO CM 


o o o 
o o m 

vO CO rH 


o o o 
o m o 

•vT rH rH 


o o o 
m o> \o 

H 


CN 

4-1 
<4-l 

•H 


o o o 


o o o 


m co cn 

odd 


•J- CM rH 

• • ■ 

O O O 


-e- M 
< 3 


ci in co 


co «<r cm 


o o o 


o o o 


o M 


CM vO CO 

<f CO en 


^O CN O 
CO CO CO 


CO CO CO 
CO CO CO 


o o o 

CO CO CO 


CO 
4-1 

e £ 

•H 
J* 


0.150 
0.140 
0.135 


0.135 
0.125 
0.120 


0.135 
0.125 
0.120 


0.135 
0.125 
0.120 


RC 

Stand. 

AASHTO 


m m o 

O Q\ o\ 

iH 


o o m 

O C\ CO 


o o m 

O CJ\ 00 

rH 


o o m 
o o\ CO 

rH 


Unified 
Classification 


Coarse Aggregates 
GW, GP 
SW, SP 


Silty Sand 

SM 


Silty Clayey Sand 
SM-SC 


Silty Clay 
CL 



CO 



6 CM 

rs 



S3 
2 



rH <T 



II II 

CO CM 
4-1 4J 
■4-1 <4-( 

a a, 

•H -H 



c 




o 




•H 




4-1 




CJ 




(0 


>> 


8- 


4-1 
•H 


o 


CO 


CJ 


c 




<D 


CD 


T3 


> 




•H 


4-1 


4-1 


,fl 


CO 


00 


rH 


•H 


CD 


CD 


pej 


rs 


II 


ii 


CJ 


6 


PS 


>- 



126 



confined modulus (i.e. C-q in Equation 9.1) and are observed to increase 
with axial stress and compaction level. Figure 9.2b shows the behavior 
of the same soil models in triaxial loading. Here, the slope of the 
curve is the tangent Young's modulus E t and are observed to decrease with 
shear stress and increase with compaction level as expected. 

The remaining three pairs of figures illustrate the same trends for 
other soil types. The silty-sand soil type (Figures 9.3a,b) have the 
largest stiffness values (slopes) in confined compression, while the 
silty-clay (Figures 9.5a,b) have the lowest. 



127 



RC = 90 




Figure 9.2a. 



8 9 10 

Percent axial strain 

Coarse Aggregates, Uniaxial Behavior 



12 



tf-ft 



- RC=105 




5 6 7 8 9 10 11 12 
Percent axial strain 

Figure 9.2b. Coarse Aggregates, Triaxial Behavior 



128 



Pa 



RCzlOO RC=90 RCi85 



£ 

4- 
(/) 

x 
< 




■» £■ 



7 8 9 10 11 12 

Percent axial strain 



Figure 9.3a Silty Sand, Uniaxial Behavior 



gj-ft 




7 8 9 10 11 12 
Percent axial strain 



Figure 9.3b Silty Sand, Triaxial Behavior 



129 



RC=90 



RC=85 




7 8 9 10 11 12 
Percent axial strain 



Figure 9.4a. Silty Clayey Sand, Uniaxial Behavior 



<fl 10 



RC-100 




Figure 9.4b, 



7 8 9 10 
Percent axial strain 

Silty Clayey Sand, Triaxial Behavior 



130 



RC= 85 




>6i 



Percent axial strain 

Figure 9.5a. Silty Clay, Uniaxial Behavior 



o;-(r 3 




Percent axial strain 

Figure 9.5b. Silty Clay, Triaxial Behavior 

131 



CHAPTER 10 

SUMMARY AND CONCLUSIONS 

This report presented a step by step development for the structural 
analysis of buried, precast reinforced concrete box culverts using the 
finite element method to model the soil-box system. Model predictions 
were validated with measured data from both in-ground and out-of-ground 
experimental tests. A user oriented soil-box model with automated finite 
element mesh generation is operational in the CANDE-1980 computer program 
and is referred to as "level 2 box" option. Also operational in CANDE-1980 
is the Duncan soil model with simplified input options for standard types 
of soil. Specific findings and conclusions from this work are listed below, 

1. Load-deformation curve predictions for reinforced concrete 
culverts are sensitive to the cracking strain parameter e . 
However at ultimate flexural capacity, the maximum stresses of 
concrete f and steel f y are the controlling parameters with 
regard to the reinforced concrete model. 

2. CANDE predictions are in good agreement with experimental data 
from reinforced concrete pipes in three-edge bearing. Better 
correlation for load-deformation curves was observed for pipes 
failing by flexure than by shear. Predicted ultimate loads, 
whether in shear or flexure, are within 10% of measured values. 

3. The measured cracking load and ultimate load for reinforced 
concrete box culverts tested in four-edge bearing show good 
correlation with CANDE predictions. Predicted cracking loads 
averaged 10% lower and ultimate loads averaged 1% lower than 
experimental results. Predictions from the SGH design/analysis 
approach are similar to CANDE predictions but showed slightly 
more deviation from experimental data. 



132 



A. Measured soil pressures from a full scale burled box installation 
are in good agreement with CANDE predictions at both intermediate 
and final burial depths. Vertical soil pressures on top and 
bottom slabs were in very close agreement, whereas measured 
lateral pressures on the sides at full burial depth were some- 
what lower than predicted. 

5. Assumed soil load distributions on buried boxes used in the 
development of ASTM C789 design tables were compared with pre- 
dicted soil load distributions determined from CANDE resulting 
in the following observations. (a) Vertical soil pressure on 
the top and bottom slabs are not uniform as assumed but increases 
monitonically from the centerline to the corners. (b) Shear 
traction on the sidewalls produces a significant downward force 
that must be equilibrated by an upward pressure on the bottom slab. 
(c) Soil stiffness is an important parameter for determining soil 
load distributions and magnitudes. The latter two effects are not 
presently taken into account in the ASTM loading assumptions. 

6. Based on CANDE predictions, the design earth covers specified in 
ASTM C789 design tables are generally conservative providing 
good quality backfill soil is assumed. However, specified earth 
covers for boxes with high levels of reinforcement tend to be 
less conservative than specified earth covers for boxes with low 
levels of reinforcement. 



133 



APPENDIX A 
DETAILS OF REINFORCED CONCRETE MODEL 

The reinforced concrete beam-rod model presented in Chapter 3 is 
discussed in further detail in this appendix. This model replaces the 
original CANDE concrete pipe type and can be used with solution levels 
1, 2 or 3. ; Subroutine CONMAT is the heart of the new reinforced concrete 
model wherein concrete cracking, loading to ultimate and unloading is 
simulated. - For purposes of this appendix, it is presumed the reader is 
familiar with basic assumptions and general solution strategy presented 
in Chapter 3. Here attention is focused on programming details. 

I. VARIABLES USED 

To calculate the initial and load dependent mechanical properties 
of the reinforced concrete sections, parameters describing the material 
behavior of concrete and steel have to be defined. Some of these para- 
meters are primary (defined by input) while others are secondary (de- 
rived from primary) . The main purpose of these parameters is to define 
an idealized stress-strain diagram for concrete and for the steel rein- 
forcement (see Figure A. 1, A. 2) . The following parameters are primary 
input data for the material properties, where, in parentheses, are the 
default values used in CANDE. 



e = concrete strain at tensile cracking (0.000 in/in) 

e = concrete strain at elastic limit (0.5 f'/E, ) 
y c 1 

e' = concrete strain at f (0.002 in/in) 
c c 

f = unconfined compressive strength of concrete (4000 psi) 
c 



E = Young's modulus for linear concrete (33 vf 1 (y) " ) 
1 c c 



134 




Strain 



Cracking 



Figure A.l - Idealized Concrete Stress-Strain Diagram. 



00 



0/ 



% 




/ 



5/ 



f 

/ 



/ 



7 

/ 



# 






Strain 



Figure A. 2 - Idealized Steel Stress-Strain Diagram. 



135 



v = Poisson's ratio for concrete (0.17) 
c 

Y = unit weight of concrete (150 pcf) 
c 

f = yield stress of steel (40000 psi) 

E = Young's modulus of steel (29 x 10 psi) 
o 

v = Poisson's ratio of steel (0.30) 
s 

The following are secondary parameters derived from primary data: 

2 
E = confined elastic modulus of concrete (E /(1-v ) 
c 1 c 

v = shear strength of concrete (2/ f ' psi) 
c c 

f = maximum tensile stress of concrete (e x E ) 

f = concrete stress at elastic limit (e x E., ) 
yc y 1 

2 
E = confined elastic modulus of steel (E /(1-v ) 
s os 

n = concrete-to-steel modulus ratio (E /E ) 

c s 

In addition, the analysis mode requires (see Figure A. 4): 
h = wall thickness of concrete (in) 



As. = area of inner reinforcement per unit length of pipe 
(in /in) 



2 



As^ = area of outer reinforcement per unit length of pipe 
(in /in) 



2 



c. = concrete cover on inner reinforcement (1.25 in) 

c = concrete cover on outer reinforcement (1.25 in) 
o 

Using these parameters the initial uncracked section properties are 
defined as: 

Effective axial stiffness: 

EA* = E (h + (n-1) As. + (n-1) As ) 
C X o 



136 



Neutral axis of bending: 

2 



_ (7- + As.(n-l)c. + As (n-l)(h-c )) 

y = E 2 x 1 o o 

c _ 

EA 
Effective bending stiffness: 

3 

EI* = E [yr + (£ - y) 2 h + (n-1) (As.(c.-y) 2 + A (h-c-y) 2 ] 

c LI I 11 so o 

The above section properties EA , y and EI are for the uncracked 
cross section of 1 inch width with no loading. Now that the initial 
stiffnesses of the beam- rod sections are defined, the beam-rod elements 
are ready to be analyzed for the first load increment. 



II. PROCEDURE TO CALCULATE THE SECTION PROPERTIES 

Using the initial section properties above, the structural system 
is solved for the first load increment resulting in trial solutions for 
thrust and moment increments within each element. Due to the non-linearity 
of the materials, the initially assumed section properties are modified 
and another trial solution is obtained. This iterative solution technique, 
which considers the average stress state during the increment to find 
effective section properties, is repeated until convergence within the 
load step is achieved. 

CONMAT subroutine evaluates EA , y, and EI for each element until 
all elements converge as the system advances from load step i-1 to load 
step i. The procedure used for each load step is as follows: 

a) For each element an increment of moment and thrust is obtained 
from the general solution process. If a section initially cracks or 
extends its crack, the stresses in the newly cracked region are zero, 

thus, the pre-existing stresses prior to cracking must be redistributed. 
This redistribution can be achieved with corrections to the thrust and 
moment increment (called here thrust and moment redistribution). In 
our approach this correction is made after the inner loop convergences, 

so that, initially, moment and thrust redistribution is zero (see step h 



137 



for redistribution). 

* _ * 

b) Assuming the section properties EA , y, and EI from the con- 
verged solution at load increment i-1 and using the increment of moment 
and thrust, a linear strain distribution for the section is calculated 
(see Figure A. 3). 

, AN , AM , -• ' 

e i = e i-i + m* + ei* (y " y) 

where 

e . 1 = strain distribution from converged solution at load 
increment i-1 
AN = increment of thrust + (thrust redistribution, last 

iteration) 
AM = increment of moment + (moment redistribution, last 
iteration) 
y = spatial coordinate from section bottom. 

This linear strain distribution e. is the first tentative solution 

1 

of the iterative procedure. 

For computational convenience, the section stiffness properties of 

EA and EI (defined in Chapter 3) are divided by the confined-elastic 

* * 
concrete modulus E , so that we may define A and I as: 

c 

it * 

A = EA /E 

* * c 
I = EI /E 
c 

Or more explicity, to obtain section properties the following integrals 

must be evaluated (see Figure A. 4): 

A = FE(y) dy + WSI (n-1) A +(WSO)(n-l) A gQ 



' o 
rh 

o 



FE(y) y dy + WSI (n-1) y . A . + WSO (n-1) y fc A 

J ' J * ti si to so 

A 



I* = FE(y)(y-y) 2 dy + WSI(n-l) (y^-y)^. + WSO (n-1 ) (y^-?)^ 



138 



s-h A £ n 



n 



6rl Si 



+ 





Figure A. 3 - Strain Distribution at Load Increment i. 



•AS, 



»AS; 



«t, 



yti 




4— X^4 



Figure A. 4 - Integration Points of the Cross Section. 



139 



where: 

FE(y) = E" /E (modulus reduction ratio for concrete) 
c c 

WSI = E' /E (for inner reinforcement reduction) 

WSO - E' /E (for -outer reinforcement reduction) 
s s ■ 

E' - tangent modulus of concrete 
c 

E' = tangent modulus of steel 
s 

Using the above formulas and the strain distribution, e . , the new 
properties EA , y and EI are calculated. From the formulas it is ob- 
served that the concrete part of the section requires an integration of 
FE(y) to evaluate its properties, where the function FE(y) is not smooth. 
Thus, Simpson's integration is performed for the concrete section using 
eleven points along its depth (see Figure A. 4). 

c) The concrete is analyzed first using the strain distribution 

e . _ and £.. The integration points are analyzed one at a time, dependent, 
in part, on the value FE(y). The first step is to update the record for 
the maximum stress-strain occurrence of each point during its loading 
history. Specifically, the converged strain distribution of load step 
i-1 and the maximum strain-stress values of the point computed in previous 
steps are used to identify if there are new maximum values to be saved. 
If the strain of the point at load step i-1 is less than any previous 
maximum strain, no change is made in the history vector. If the strain 
of the point at load step i-1 is greater than the maximum strain, a new 
maximum was reached for that point at load step i-1. Using the new 
maximum strain, the new maximum stress is computed using the old maximum 
stress-strain value located along the basic stress-strain diagram (see 
Figure A.l). The value of the maximum stress-strain of the point is 
saved and subsequently used to define the unloading and reloading path 
of the stress-strain diagram at load step i. 

d) A comparison of the strain e . with e . 1 for the integration 
point indicates if the point is loading or unloading. If the point is 
unloading ( |e| < | e J ) the stress a of the point can be calculated 



140 



knowing that the unloading is elastic, 

a . = o + (e. - e )E 
1 max 1 max c 

where (a , e ) = maximum stress-strain value 
max max 

e . = strain of the point at load step i 

If the point was not previously cracked and a. doesn't reach the 
concrete tensile strength then FE(y) = 1.0, if the point was previously 
cracked and a. is in compression then FE(y) = 1.0. Otherwise FE(y) is 
0.0 (see Figure A. 5). 

If the point is loaded, it can be initial loading or reloading. 

Using the previous strain-stress values (e . , a. ) and the maximum 

strain-stress of the point (e , a ) , the loading or reloading cases 

max max 

are identified. If the previous values are equal to the maximum it 

means initial loading, otherwise it means reloading. Once the case is 

defined and the values of (e . ,, a. ,) and (e , a ) are known, the 

x-1 l-l max max 

case falls in one of the possible point histories presented in Figure A. 5. 

Knowing in what case the point is, the stress value a. of the point for 

load step i is evaluated. Now that the stress-strain value (a., e.) is 

11 

defined. E' is determined by the slope between (e . n , a. .) and (e . , a.). 

c l-i 1-1 1 1 

Thus FE(y) for each point is: 

1 1-1 c 

This procedure is executed until FE(y) is defined for the eleven inte- 
gration points. 

e) In the previous step (d), when the value of FE(y) is determined 
for each integration point, at the same time, it is possible to determine 
what points are cracked or uncracked. With this data and doing a linear 
interpolation of stresses between the points of crack and no crack, the 
crack depth of the section is calculated. This calculated crack depth 
is printed out. Note crack depth is completely different than crack 
width discussed in Chapter 3. 



141 







7 




>S 








* (e^* cr^) load step i-1 | FE(I) 

X ^ C i* °i^ load step i FE(I) 

Ei I 

s tangent modulus i 

\= maximu m strain-stress history 



E (Z^ 

0.0 (vhen cracked and 
a. less than zero) 



Figure A. 5 - Modulus Function for all Possible Concrete Strain Histories 
at a Point in the Beam Cross Section. 



- 142 



f) Knowing the cross section strain distribution £•_-]» e - an ^ 
the location of the reinforcement in the cross section, the strains of 
the inner and outer reinforcement are calculated for load step i-1 and 
i. Using the stress-strain values at load step i-1 and the strain value 
at load step i for the reinforcement, the value of the stress at load 
step i is calculated for elastic loading or unloading as: 

o . = a . - + (e - e . ) E 
i l-l 1 i-I s 

For plastic loading we have: 

a. = f (steel yielding stress) 

This procedure applies to both inner and outer reinforcement. Using 
their stress-strain values, the factors WSI and WSO for the inner and 
outer reinforcement are calculated respectively as follows : 

(a i ~ a i-l } 
WSI = , r (stress-strain values for the inner 



< e i " E i-1> E s 



reinforcement ) 



( °i " ? i-l ) 
WSO = — ( r (stress-strain values for the outer 

i i-l s reinforcement) 

All the possible cases of strain histories for the steel reinforcement 
are shown in Figure A. 6. 

g) With the preceeding developments, the section properties defined 
in step b are evaluated with the aid of eleven point Simpson integration 
as follows: 

A = SUM1 + SI + SO 

- = (SUM2 + y. SI + y SO)/A 
y xo 

i* = y 2 sumi - 2y sum2 + SUM3 + (y^-y) 2 si + (y Q -y) SO 

In the above, SUMI, SUM2, and SUM3 represent the concrete contri- 
butions and are the numerical integrations of the integrands FE(y), 

2 
yFE(y) and y FE(y), respectively, i.e., 

SUMI - 4? (FE(1) + 4FE(2) + 2FE(3) + ••• FE(ll)) 



143 






(£. ,, "O^i) l° a <i step i-1 
X (Sj, <7.) load step i 



Ee= steel Young's modulus 



E* = tangent modulus 

wsi(o) - eVe_ 



Figure A. 6 - Modulus Function for all Possible Steel Strain Histories 
at a Point in the Beam Cross Section. 



144 



SUM2 = 4^ (FE(l)y + 4FE(2)y 2 + 2FE(3)y 3 + ••• FE(ll)y ) 

SUM3 = 4f (FE(l)y? + 4FE(2)yJ + 2FE(3)y^ + ••• FE(ll)y^) 

Steel contributions SI and SO are associated with inner and outer 
reinforcement and are given by: 

SI = WSI(n-l)A . 
si 

SO = WSO(n-l)A 

so 

If either or both reinforcements are located in a cracked zone, the value 
of "n" is used instead of "n-1". That is, n-1 accounts for the reinforce- 
ment hole in uncracked concrete. 

* _ * 

These values of A , y and I are compared with the assumed values 

at step (b). The value A is used to check convergence. If the con- 
vergence is not reached, a new set of strains e. are evaluated at step 
(b) using the new A , y, I . The entire procedure is repeated until 

•A* mm J- 

successive values of A , y, and I converge or after four iterations 
(inner loop), then the program goes to the next step h. 

h) Now that new values of A , y and I are known for the cross 
section at load step i (however, convergence of AM and AN are not yet 
assured), a modified set of strain-stress values (e-. , a.) for each 
point in the cross section is computed to account for stress redistri- 
bution due to cracking. These values are compared with the strain- 
stress value (e. ,, a. ,). If for some point the stress a. implies 
l-l l-l i 

that the point is cracked and was not previously cracked, it means 
that the stress a. must be redistributed to the remaining uncracked 
concrete and reinforcement. The procedure adopted in the program is 
to evaluate an equivalent moment and thrust to be redistributed due to 
the cracking of those points at load step i, where: 

AN = Z Ay • a (thrust redistribution) 

AM^ = Z Ay * o (y-y) (moment redistribution) 



145 



After the moment and thrust redistribution is complete the program is 
ready to go to step (a) and do the same procedure for the next element 
(node) . 

i) All elements of the reinforced concrete structure are analyzed 

4c — 4c 

and the values of A , y, I , AN R and AM are calculated for each one. 

4c ~ 

If the values of A and y obtained for every element converge with the 

ones obtained in the previous iteration and AN , AM are zero for all 

R R 

the nodes, the trial solution has converged. Otherwise, the system is 

* °" 4c 

solved again using the values of A , y and I from the last iteration. 
This is called "inner loop" iteration. Once the inner loop converges, 
the section properties, EA , y, EI , are used to get another solution to 
the entire soil-structure system. This gives new values for AM and AN 
to repeat the inner loop. Successive solutions for AM and AN is called 
outer loop iteration. 

Outer loop iteration is repeated until a convergence is reached 
between successive solutions for AM and AN. If convergence is not 
reached after six trial response solutions, the program assumes the 
last one as an approximate solution and advances to next load step. 



146 



APPENDIX B 

CANDE-1980: USER MANUAL SUPPLEMENT 

The following user's guide is a supplement to the 1976 "CANDE USER 
MANUAL" (2). This supplement provides input instructions for the new 
options on reinforced concrete box culverts and Duncan soil model described 
in the main body of this report. 

The original 1976 manual is still the principal reference source and 
may be used without reference to this supplement if the new options are 
not desired. Taken together, the original and new options form the program 
called CANDE-1980. Input instructions for CANDE-1980 follows the same 
pattern as the original program, composed of three main sections (A,B, and 
C) as shown in Figure B.l, where new options are marked with an asterisk. 

Section A is the master control input (card 1A) and is unchanged from 
the original program. Section B (cards IB to 3B) includes a new input 
option for modeling reinforced "concrete box" culverts in addition to 
the original pipe types. The "concrete box" culvert type is only operative 
in the analysis mode and cannot be used with solution level 1. Section C 
(card sets C and D) includes the new "level 2 box" finite element generation 
scheme discussed in Chapter 6 along with the new Duncan soil model option 
presented in Chapter 9. 

The supplemental input instructions to be given here provide a complete 
set of data input for the subset of options shown in Figure B.2. Thus, 
these instructions are self contained for "level 2 box" solutions with any 
soil model. For extended level 2 and level 3 options, however, the 1976 
user manual must also be used where noted. 

Formatted input instructions for Sections A, B, and C are presented 
in order, followed by explanatory comments and illustrations. Example 
input-output data is given in the next appendix. 



147 



SECTION A - MASTER CONTROL: CARD 1A 



• Execution mode = design or analysis 

• Culvert (pipe) type ■ steel, etc. 

• Solution Level = 1,2, or 3 (option, level 2-extended) 



SECTION B - CULVERT TYPE INPUT: CARDS IB up to 3B 



Steel 



1B,2B 



Aluminum 



1B,2B 



Concrete 
Pipe 

1B,2B,3B 



Plastic 



1B,2B 



Basic 



1B,2B 



Concrete 
box 

1B,2B,3B 



SECTION C - SOLUTION LEVEL INPUT: CARDS CI up to C7 



Level-1 



1C,2C 



Level- 2 
pipe 

1C,2C,3C 



Level-2 
box* 

1C,2C 



Level-2 
extended 

4C to 7C 



Level-3 



1C to 5C 



"V 



(End) 







i 








SOIL MODELS: CARDS Dl up to D4 


Elastic 
ID, 2D 


Ortho 
Elastic 

ID, 2D 


it 

Duncan 
model 

ID to 4D 


Over- 
Burden 

ID, 2D 


Hardin 
model 

ID, 2D 


Inter- 
face 

ID, 2D 




* New CANDE Options. 



FIGURE B.l General Input Flow for CANDE-1980 



148 



SECTION A - MASTER CONTROL: CARD 1A 

• Execution mode = analysis 

• Culvert type = concrete (box) 

• Solution level = 2 



SECTION B - CONCRETE BOX INPUT: CARDS IB to 3B 

• Reinforced concrete material properties (IB, 2B) 

• Option for standard cross-sections (3B-1, 3B-2) 

• Option for arbitrary cross-sections (3B) 






SECTION C - LEVEL 2- BOX: CARDS 1C, 2C, ID to 4D 

• Installation type (1C) 

• Box-soil dimensions (2C) 

• Soil model type, density (ID) 

- Elastic (2D) 

- Orthotropic elastic (2D) 

- Duncan model (2D to 4D) 

- Overburden dependent (2D) 

- Hardin model (2D) 

- Interface model (2D) 



FIGURE B.2 




Specific input flow for level-2 box culverts 
described in supplemental user's manual. 



149 



SECTION A - MASTER CONTROL CARD 

Input 

Card 1A. Master control card (one card per problem) : 



Columns 
(format) 

01-06 
(A4,2X) 



08-08 
(ID 



Variable 
(units) 

XMODE 
(word) 



LEVEL 



Entry Description 

Word defining program mode, 

= ANALYS, denotes analysis problem 
= STOP, program terminates, last 
card in deck 

Defines solution level to be used 
= 2, denotes finite element 

solution with automated mesh 
= 3, denotes finite element 

solution with user-defined mesh 



Notes 



(1) 



(2) 



10-15 


PTYPE 


(A4,2X) 


(word) 


17-76 


HED 


(15A4) 


(words) 


77-78 


NPMAT 


(12) 




79-80 


NPPT 


(12) 





Defines pipe material to be used, 
= CONCRE, denotes reinforced 
concrete (pipe or box) 

User defined heading of problem to be 
printed with output 

Number of pipe elements; only required 
when LEVEL = 3 

Number of pipe nodes; only required 
when LEVEL = 3 



(3) 



*** GO TO SECTION B *** 



150 



SECTION B - BOX CULVERT 



Reinforced Concrete Input (Cards IB, 2B, 3B) 

Card IB. Material properties of reinforced concrete. 



Columns 


Variable 


(format) 


(units) 


01-10 


PDIA 


(F10.0) 


(in.) 


11-20 


PT 


(F10.0) 


(in.) 


22-25 


RSHAPE 


(A4) 


(word) 



Entry Description 



Notes 



26-30 
(15) 



NONLIN 



31-40 


STNMAT 


(1) 


(F10.0) 


in/in. 




41-50 


STNMAT 


(2) 


(F10.0) 


in. /in. 




51-60 


STNMAT 


(3) 


(F10.0) 


in/in. 





Any negative value. This signals (4) 
program that is working with box culverts 

Nominal concrete wall thickness. This (5) 
value is used whenever wall thickness 
is not specified on Card 3B 

Control word to. select manner of (6) 

input for cross section properties 
on Card 3B 

= ARBI, implies section properties 

at each node along connected 

sequence will be specified by 

user (arbitrary) 
= STD, implies simplified property 

input will be allowed in conjunction 

with Level 2 box mesh 

Degree of nonlinearity , (7) 

■ 1, concrete cracking only 
= 2, also include nonlinear 

compression of concrete 
= 3, also include steel yielding 

Concrete strain at which tensile (8) 

cracking occurs (positive), 
Default =0.0 in/in. 

Concrete strain at elastic limit 
in compression (positive) 

Default = 1/2 PFPC/PCE (see next card) 

Concrete strain at initial 
compressive strength, f^, (positive), 
Default = 0.002 in/in. 



151 



Card 2B. 


Concrete and 


Columns 


Variable 


(format) 


(units) 


01-10 


PFPC 


(F10.0) 


(psi) 


11-20 


PCE 


(F10.0) 


(psi) 


21-30 


PNU 


(F10.0) 




31-40 


PDEN 


(F10.0) 


(pcf) 



41-50 


PFSY 


(F10.0) 


(psi) 


51-60 


PSE 


(F10.0) 


(psi) 


61-70 


PSNU 


(F10.0) 




71-80 


SL 


(F10.0) 


(in.) 



steel properties: 



Entry Description 

Compressive strength of concrete, f£, 
Default - 4,000 psi 

Young's modulus of concrete in 

elastic range, ~.~ . , ? 

Default = 33 (density) ' (f) ' 

c 

Poisson ratio of concrete, 
Default = 0.17 



Unit weight of concrete (density) 
Default = 0.0 for body weight; 
however, 

Default = 150 pcf for modulus 
calculation 

Yield stress of reinforcing steel, 
Default = 40,000 psi 

Young's modulus of steel 
Default = 29 x 10 6 psi 

Poisson' s ratio of steel, 
Default =0.3 

Spacing of reinforcement 
Default = 2.0 in. 



Notes 
(8) 



(9) 



(10) 



152 



Card 3B. For RSHAPE = STD (only), two cards are required. 
Only for Level 2 Box. 



Card 3B-1. 


Concrete 


Box-wall dimensions 


Columns 


Variable 


Entry Description 


01-10 


PTT 


Thickness of top slab 


(F10.0) 


(in.) 


Default = PT 


11-20 


PTS 


Thickness of side slab 


(F10.0) 


(in.) 


Default = PT 


21-30 


PTB 


Thickness of bottom slab 


(F10.0) 


(in.) 


Default - PT 


31-40 


HH 


Horizontal haunch dimens 


(F10.0) 


(in.) 




41-50 


HV 


Vertical haunch dimensio 


(FIO.O) 


(in.) 





(Notes) 
(11) 



(12) 



Card 3B-2. 


Steel rei 


Columns 


Variable 


01-10 
(F10.0) 


AS1 
(in.2/in.) 


11-20 
(F10.0) 


AS 2 
(in.2/in.) 


21-30 

(F10.0) 


AS 3 
(in.2/in.) 


31-40 
(F10.0) 


AS4 
(in. 2/in. ) 


41-50 
(F10.0) 


XL1 


51-60 
(F10.0) 


TC 
(in.) 



Entry Description 

Outer steel area, side wall 

Inner steel area, top slab 

Inner steel area, bottom slab 

Inner steel area, side wall 



Length ratio of AS1 steel along 
top (bottom) slab 

Uniform thickness of cover to all 
steel centers 

Default = 1.25 in. 



(13) 



*** GO TO SECTION C *** 



153 



Card 3B. For RSHAPE = ARBI (only) repeat this card for 
number of pipe nodes (NPPT*) . This card may 
be used for Level 2 or Level 3. 



Entry Description Notes 

Area of inner steel reinforcement (14) 

Area of outer steel reinforcement 



Thickness of concrete cover to 
center of inner steel 
Default = 1.25 in. 

Thickness of concrete cover to 
center of outer steel 
Default = 1.25 in. 

Thickness of concrete 
Default = PT 



* In Level 2 solution NPPT = 15 



*** GO TO SECTION C *** 



Columns 
(format) 


Variable 
(units) 


01-10 
(F10.0) 


AS I 
(in.2/in. ) 


11-20 
(F10.0) 


ASO 

(in. 2/in. ) 


21-30 
(F10.0) 


TBI 
(in.) 


31-40 
(F10.0) 


TBO 
(in.) 


41-50 
(F10.0) 


PTV 
(in.) 



154 



Columns 


Variable 


(format) 


(units) 


01-04 


WORD 


(A4) 





SECTION C - SOLUTION LEVEL DESCRIPTION 

Level 2 Input (Cards 1C, 2C, ID, 2D) 

Card 1C. Define mesh type, title, and special options: 

Entry Description 



Name to identify type of automatic 
mesh 

= EMBA, embankment mesh 

- TREN, trench mesh 

User description of mesh to be 
printed with output 

Command to permit user to selectively 

modify the automatic mesh, 

= MOD, mesh will be modified 
^ MOD, mesh will not be modified 
(left, justified) 



05-72 
(17A4) 

73-76 
(A4) 



TITLE 



W0RD1 



Notes 
(15) 



(16) 



For level 3 input, see Section C in 1976 manual. 



155 



Card 2C. Define print options and mesh parameters: 



Columns 
(format) 

01-05 
(15) 



06-10 
(15) 



11-15 
(15) 



16-20 
(15) 



Variable 
(units) 

IPLOT 



IWRT 



MGENPR 



NINC 



21-30 


Rl 


(F10.0) 


(in.) 


31-40 


R2 


(F10.0) 


(in.) 


41-50 


HTCOVR 


(F10.0) 


(ft.) 


51-60 


DENSTY 


(F10.0) 


(pcf) 



Entry Description Notes 

Signal to create a plot data tape 
on unit 10 

= 0, no data tape created 

= 1, create data tape 

Signal to print out soil response 
for all elements, 

= 0, no soil response printed out 

= 1, print out soil response 

Code to control amount of print out 
of mesh data, 

= 1, minimal printout; just 

control data 
= 2, above, plus node and element 

input data 
= 3, above, plus generated mesh data 
= 4, maximal printout of input data 
Default = 3 

Number of construction increments, (17) 
= -1, combine all lifts into 

one monolith 
= 0, used for data check only; 

all data is read but not executed 
= N, number of construction 

increments to be executed, 

N = 1 to 20 

Distance from center of the box to (18) 
center of side wall 

Half of the distance from center of top 
slab to center of bottom slab. 

Height of soil cover over the top (19) 
of the box 

Density of soil above truncated mesh 
to be used as equivalent overburden 
pressure 



continued 



156 



Card 2C. 


continued 


Columns 


Variable 


(format) 


(units) 


61-70 


TRWID 


(F10.0) 


(ft.) 


71-80 


BDEPTH 


(F10.0) 


(in.) 



Entry Description 

Width of trench; only required 
for WORD = TREN 

Depth of bedding material 
Default = 12 in. 



Notes 
(20) 

(21) 



For extended level 2 option (W0RD1=M0D) , insert cards 4C to 7C here prior 
to card set D (See note 16). Otherwise, go directly to card set D. 



157 



Soil Data Cards: 

Card ID. Material identifier card (repeat D cards for each material) 



Columns 
(format) 

01-01 
(Al) 



02-05 
(14) 



06-10 
(15) 



11-20 
(F10.0) 



21-40 
(5A4) 



Variable 
(units) 

LIMIT 



ITYP 



DEN (I) 
(pcf) 



MATNAM 
(words) 



Entry Description Notes 

Last material card-set indicator; 
= 0, read another set of material 

definitions 
= L, this is the last material 

input 

Material zone identification number (22) 
for level 2 box use: 

= 1, for in-situ soil zones 

= 2, for bedding zones 

= 3, for fill soil zones 

Selection of material model to be (23) 
associated with material zone I, 

= 1, linear elastic (isotropic) 

= 2, linear elastic (orthotropic) 

= 3, Duncan soil model 

= 4, overburden dependent model 

= 5, Hardin soil model 

= 6, frictional interface 

(not operative with Level 2 box) 

Density of material I used to compute 
gravity loads; not applicable for 
ITYP = 6. 

For ITYP = 3,4, or 5, MATNAM is used (24) 
to select soil subgroup models as shown 
in Table B.l on the next page. In all 
cases, MATNAM is printed out with the 
data but has no control for ITYP = 1, 2 
or 6. 



Go to card 2D corresponding to ITYP. 



158 



TABLE B.l. Soil models controlled by MATNAM (Card ID) 



MATNAM 


Soil Model Description 






ITYP = 3, Duncan soil model, Chapter 9, Table 9.2 


CA105 

CA95 

CA90 

SM100 

SM90 

SM85 

SC100 

SC90 

SC85 

CL100 

CL90 

CL85 

USER 


coarse aggregate, relative compaction 105% 
coarse aggregate, relative compaction 95% 
coarse aggregate, relative compaction 90% 
silty sand, relative compaction 100% 
silty sand, relative compaction 90% 
silty sand, relative compaction 85% 
silty clayey sand, relative compaction 100% 
silty clayey sand, relative compaction 90% 
silty clayey sand, relative compaction 85% 
clay, relative compaction 100% 
clay, relative compaction 90% 
clay, relative compaction 85% 
Parameters supplied by user 


ITYP = 4, Overburden Dependent, 1976 CANDE manual, pg. 39 


GGOOD 
GFAIR 
MGOOD 
MFAIR 
CGOOD 
CFAIR 
USER 


granular soil, good compaction 
granular soil, fair compaction 
mixed soil, good compaction 
mixed soil, fair compaction 
cohesive soil, good compaction 
cohesive soil, fair compaction 
parameters supplied by user 


ITYP = 5, Hardin soil model, 1976 CANDE manual 


GRAN 
MIXED 
COHE 
TRIA 


granular soil, specified void ratio 
mixed soil, specified void ratio 
cohesive soil, specified void ratio 
parameters specified by user (triaxial test) 



* MATNAM must be left justified (i.e. start in column 21) 
Defaults are: MATNAM = USER for ITYP = 3 and 4, or 
MATNAM = MIXED for ITYP = 5. 



159 



Card 2D. ITYP = 1, linear elastic- 



Columns 
(format) 

01-10 
(F10.0) 

11-20 
(F10.0) 



Variable 
(units) 

E 
(psi) 

GNU 



Entry Description 

Young's modulus of materil I 

Poisson's ratio of material I 



Notes 
(25) 



Card 2D, ITYP = 2, orthotropic, linear elastic: 

Columns Variable 

(format) (units) Entry Description 

01-10 CP(1,1) Constitutive parameter at matrix 

(F10.0) (psi) position (1,1) 

11-20 CP(1,2) Constitutive parameter at matrix 

(F10.0) (psi) position (1,2) 

21-30 CP(2,2) Constitutive parameter at matrix 

(F10.0) (psi) position (2,2) 

31-40 CP(3,3) Constitutive parameter at matrix 

(F10.0) (psi) position (3,3) 

41-50 THETA Angle of the material axis with 

(F10.0) (deg) respect to the global x-axis 



Notes 
(26) 



Card 2D, ITYP = 3, Duncan soil model 



Columns 
(format) 

01-05 
(15) 

06-15 



Variable 
(units) 

NON 



RATIO 



Entry Description 

Maximum number of iterations 
Default = 5 

Moduli averaging ratio 
Default =0.5 



Notes 
(27) 

(28) 



Go to cards 3D and 4D if MATNAM = USER. Otherwise input is complete for 
Duncan model. 



160 



Card 3D, Hyperbolic parameter for tangent Young's modulus 



Columns 


Variable 


(format) 


(units) 


01-10 


C 


(F10.0) 


(psi) 


11-20 


PHIO 


(F10.0) 


(radians) 


21-30 


DPHI 


(F10.0) 


(radians) 


31-40 


ZK 


(F10.0) 




41-50 


ZN 


(F10.0) 




51-60 


RF 


(F10.0) 





Entry Description Notes 

Cohesion intercept (29) 

Initial friction angle 

Reduction in friction angle for 
a 10-fold increase in confining 
pressure 

Modulus number, K 
Modulus exponent, N 
Failure ratio, R_ 



Card 4D, Hyperbolic parameters for tangent bulk modulus, or constant 
Poisson ratio option. 



Entry Description Notes 

Bulk modulus number, K, (30) 

Bulk modulus number, M 



Poisson 1 s ratio. If a nonzero value 
is entered, the bulk modulus is not 
used. Instead, the specified constant 
VT is used. 



Columns 


Variable 


(format) 


(units) 


01-10 


BK 


(F10.0) 




11-20 


BM 


(F10.0) 




21-30 


VT 


(F10.0) 





161 

i 



i 



Card 2D, ITYP = 4 (MATNAM = USER), Overburden dependent model, 
user defined table, repeat Card 2D as needed to define 
input table, last card is blank to terminate reading. 



Columns 


Variable 


(format) 


(units) 


01-10 


H(N) 


(F10.0) 


(psi) 


11-20 


E(N) 


(F10.0) 


(psi) 


21-30 


GNV(N) 


(F10.0) 





Entry Description 

Overburden pressure for 
table entry N 

Young's secant modulus for 
table entry N 

Poisson's ratio table entry N 



Notes 
(31) 



* Note, Card 2D is not required if MATNAT is other than USER since 
overburden dependent tables are stored in CANDE for specified 
categories of soil. 



ITYP =5, and MATNAM = GRAN, MIXE, or COHE; Extended-Hardin 



Card 2D. 


ITYP = 5, a 




model for t 


Columns 


Variable 


(format) 


(units) 


01-10 


XNUMIN 


(F10.0) 




11-20 


XNUMAX 


(F10.0) 




21-30 


XQ 


(F10.0) 




31-40 


V0IDR 


(F10.0) 




41-50 


SAT 


(F10.0) 




51-60 


PI 


(F10.0) 





Entry Description 

Poisson's ratio at low shear strain 
Default =0.10 

Poisson's ratio at high shear strain 
Default =0.49 

Shape parameter q for Poisson's ratio 
function 

Default =0.26 

Void ratio of soil, range 0.1 to 3.0 



Ratio of saturation, range 0.0 to 1.0 



Plasticity-index/ 100, range 0.0 to 1.0 



Notes 
(32) 



61-65 



NON 



Maximum iterations per load step; 
Default = 5 



162 



Card 2D. ITYP ■ 5, and MATNAM = TRIA; Extended-Hardin model for 
triaxial data input 



Entry Description 

Same as card above (XNUMIN, XNUMAX, XQ) 

Hardin parameter used to calculate 
maximum shear modulus 

Hardin parameter used to calculate 
reference shear strain 

Hardin parameter used to calculate 
hyperbolic shear strain 

Maximum iterations per load step 
Default = 5 



Columms 


Variable 


(format) 


(units) 


01-30 




31-40 


SI 


(F10.0) 




41-50 


CI 


(F10.0) 




51-60 


A 


(F10.0) 




61-65 


NON 


(15) 





Card 2D. ITYP = 6, interface property definition 



Columns 


Variabl 


(format) 


(units) 


01-10 


ANGLE 


(F10.0) 


(deg) 


11-20 


FC0EF 


(F10.0) 




21-30 


TENSIL 


(F10.0) 


(lb/in) 



Entry Description Notes 

Angle from x-axis to normal (33) 

of interface 

Coefficient of friction 



Tensile breaking force of contact 
nodes 



* * * End of input * * * 



163 



COMMENTARY NOTES 

(1) Each problem begins with the command ANALYSIS. The DESIGN option 
is not available for box culverts. The program will continue to 
execute problem data sets back-to-back until the command STOP is 
encountered. 

(2) Setting LEVEL=2 signals the program that the automatic mesh 
generation feature will be used. "Level 2 box" is distinguished 
from "Level 2 pipe" by a subsequent instruction in Section B. 
Setting LEVEL=3 allows description of arbitrary reinforced concrete 
structures and loading conditions. LEVEL=1 is not operable for 
box cuJ verts. 

(3) By setting PTYPE = CONCRE, the reinforced concrete beam-rod element 
is used to model the culvert (for other pipe types see 1976 manual). 

(4) Setting PDIA = -1.0 signals the program that Section B input data 
is for a concrete box as opposed to a concrete pipe. Also, if 
LEVEL = 2, it subsequently signals the program to read Section C 
input for "level 2 box" instead of "level 2 pipe". 

(5) Defining the default concrete wall thickness, PT, is simply for 
convenience in limiting input data on subsequent cards. 

(6) RSHAPE controls the two options for defining section properties 
around the box. Setting RSHAPE = STD allows simplified input 

for standard ASTM box sections and can only be used with LEVEL = 2. 
Setting RSHAPE = ARBI allows the user to arbitrarily define section 
properties at each node around the box and may be used with LEVEL = 
2 or 3. 

(7) Generally set NONLIN = 3 for all problems. Other options are pri- 
marily for behavior studies. 

(8) Figure B.3 illustrates the concrete material parameters representing 
the concrete stress-strain behavior. Generally, the default options 
provide reasonable parameter values except for cracking strain 



164 



STNMAT(l) and compressive strength PFPC which are conservative. 
Cracking strain values up to 0.0001 were used for box culverts 
studied in this report. 

(9) Figure B.4 illustrates the reinforcement material parameters 
representing the steel stress-strain behavior. Default options 
provide reasonable parameter values except for steel yield stress 
which is conservative. Standard ASTM box section reinforcement 
assumes 65,000 psi yield strength. 

(10) The spacing parameter is used only for crackwidth predictions in 
the Gergely-Lutz formula (see Chapter 3). The default value was 
used in this study. 

(11) For the RSHAPE = STD option, refer to Figure B.5 for illustration 
of standard box section parameters. If the top, side and bottom 
slabs are the same thickness, these input variables can be skipped 
and the default value PT, input on Card IB, will be used. 

(12) Haunch dimensions are used by CANDE to increase the wall thickness 
at corner nodes by a simple averaging process and are shown on 
the printed output. Generally, HH = HV = PT. 

(13) Steel placement is illustrated in Figure B.5 and corresponds to 
standard ASTM box designs. All reinforcement steel areas are to 
be defined per inch of length in the longitudinal direction. Con- 
crete cover to all steel centers is specified with the parameters 
TC. If variable TC values are desired use RSHAPE = ARBI. 

(14) For the RSHAPE = ARBI option, refer to Figure B.6 for illustration 
of parameters. In the level 2 option, the section properties are 
defined individually at the 15 points (nodes) shown in the figure. 
For level 3 solutions, the section properties are defined at the 
culvert nodes (NPPT) established by the user. 

(15) The embankment and trench configurations are illustrated in 
Figure B. 7 and B.8. Each is composed of three soil zones; in situ, 
bedding, and fill. 



165 



(16) By setting W0RD1 = MOD, the level 2 mesh can be selectively 
modified by using the extended level 2 option. Modifications 
include; defining new soil zones and shapes and specifying live 
loads. When this option is exercised, additional data cards C4 to 
C7 are inserted after card C2. Card C3 does exist for level 2 
box input. Input instructions for cards C4 to C7 are in the 1976 
user manual and the finite element mesh topology for level 2 box 
is shown in Figure B.ll to B.14. 

(17) Construction increments for the trench and embankment installations 
are shown in Figures B.9 and B.10. In both cases, the first con- 
struction increment contains the box culvert and in situ soil. 
Increments 2,3,4 are each composed of two rows of elements uni- 
formly spaced along the sides of the box. Increments 5 to 9 are 
composed of one element row, increment 5 is 1/3 R2 thick, and 
increments 6 to 9 are 2/3 R2 thick. For deep fill heights, sub- 
sequent increments are formed with equivalent overburden pressure 
(see note 19). The special case of NINC = -1 combines all incre- 
ments into one (not recommended). 

(18) Rl and R2 define the box size and control the overall dimensions 
of the mesh as shown in Figure B.ll. 

(19) HTCOVR is the distance from the middle of the top slab to the 
final soil surface. If HTCOVR is specified greater than 3R2, 
the mesh top boundary is truncated at the 3R2 level, and the 
remaining soil load is applied as equivalent increments of over- 
burden pressure (i.e. DENSTY * (HTCOVR - 3R2)/(NINC - 9)). If 
HTCOVR is specified less than 3R2, the horizontal mesh line 
closest to HTCOVR is moved to the specified height, but with the 
condition that at least two layers of soil exist over the top 

of the box. 

(20) TRNWID defines the trench width from the middle of the box sidewall 
to the in situ soil as shown in Figure B.7. The vertical mesh 
line (Figure B.ll) closest to the specified position is moved to 
this position to form the trench wall boundary. Minimum value 



166 



for TRNWID is 0.1 Rl. If TRNWID is greater than 4R1 an embankment 
installation is obtained. 

(21) BDEPTH defines the depth of bedding below the bottom slab. The 
bedding zone is composed of one layer of elements and is kept within 
the depth limits (1/10) R2 to (2/3)R2. The bedding width extends 
one element beyond the box side. 

(22) For level 2 box, three sets of D cards are to be input corresponding 
to the predefined soil zones: I = 1, 2, and 3 implying in situ, 
bedding, and fill, respectively. For level 3, I corresponds to 
material number of element defined by user. 

(23) Any soil model (ITYP = 1, 2, 3, 4 or 5) may be assigned to any 
soil zone. Choice of a soil model is dependent on the problem 
objective, availability of actual soil data, and user's preference. 
Suggestions for soil model applications are given in subsequent 
notes. 

(24) The MATNAM subcategories provide a simplified data input option 
for ITYP = 3, 4, and 5 wherein the soil model parameters for 
standard types of soil are stored in the CANDE program. Alternatively, 
by setting MATNAM = USER (or MATNAM = TRIA for ITYPE = 5) model 
parameters may be defined by the user. 

(25) The linear elastic model (ITYP = 1) is useful for parameter studies 
and bracketing solutions with soft and stiff moduli values. See 
Chapter 7 (Table 7.2) for typical moduli values. It is generally 
reasonably to model in situ soil with the elastic model. 

(26) Orthotropic models can be used to simulate reinforced earth (see 
Reference 31). 

(27) If the maximum number of iterations (NON) for convergence is ex- 
ceeded, the program advances to the next load step. If NON is 
specified as a negative value, iteration values and convergence 
checks are printed out. 

(28) Generally set RATIO = 0.5. If Duncan model is used for pre-existing 
soil zones (e.g. bedding and in situ), set RATIO = 1.0. The Duncan 



167 



model is probably best suited for characterizing fill soil. See 
Chapter 9 (Table 9.2) for hyperbolic parameters corresponding to 
standard soil types stored in the program. 

(29) For MATNAM = USER, the tangent Young's modulus hyperbolic para- 
meters are input by the user (usually determined from triaxial 
tests, see Reference 29). 

(30) For MATNAM = USER, the tangent bulk modulus hyperbolic parameters 
may be specified. Or, as an alternative, a constant Poisson's 
ratio may be specified. The latter option is the original version 
of the Duncan soil model, still preferred by some investigators. 

(31) Moduli values for the overburden dependent model correspond to 
secant relations from confined compression tests. Thus, this 
model provides reasonable representation of soil behavior in 
zones where deformation is primarily vertical. For this reason, 
the model is better suited for rigid culvert installations than 
flexible culvert installations. If MATNAM is other than USER, the 
table entries are automatically supplied by CANDE. Table values 
are listed in the 1976 user manual, page 39. 

(32) The Hardin model is discussed in detail in the 1976 CANDE manual. 
This option is best utilized in conjunction with triaxial test 
data. 

(33) See the CANDE 1976 manual. 



168 



nitia' crushing 




tensile njpture 



Flgufie B.3 - Idealized Stress-Strain Diagram of Concrete. 



f« 



PFSY 




->S S 



Figure B.4 -- Idealized Stress-Strain Diagram of Steel. 



169 



AS2 



HH 



2 R2 



Hvt 



AS3 / t 
L 2_£1 



LI 



\ 


/— 


t 

PTT 


\, . 


- r-TC 

<-PTS 




AS1_^" 


\_ 


PTB 


/ 




r 





AS4 



XL1 = 



JJ. 
R1 



Figure B.5 - Box Culverts Parameters for RSHAPE - STD 
and Level 2 Solution. 



2 R2 



2 



3 



4 5 

9 • 



( 

\ 



+6 

7) 

.8 



15 14 13 12 11 

R1 L 



TBI 



A 



/ 



Asiy 



k 



K 






aso 



'BO 



PTV 



+ 



Figure B.6 - Box Culvert Generated Mesh (Level 2), and Properties 
Definition (Level 2 or 3) . 



170 




equivalent 
overburden 
soil pressure 



3R2 



2R2 



3R2 



Figure B.7 - Trench Soil Installation. 




3R2 



2R2 



/ f equivalent 

overburden 
^ soil pressure 



3R2 



Figure B.8 - Embankment Soil Installation. 



171 



3R2 

R2 
R2 

3R2 



truncated soil equivalent 
jS over burden pressure (n-9) 



V y V 




Figure B.9 - Soil Layers and Construction Increment when they 
are Applied for Trench Soil Installation. 



truncated soil equivalent 
^x" over burden pressure (n-9) 



3R2 

R2 

R2I 

3R2 




Figure B.10 -Soil Layers and Construction Increment when they 
are Applied for Embankment Soil Installation. 



172 



|R2 
fR2 
•|R2 



•§R2 



R2 



R2 



R2 



|R2 
§R2 



¥ 2 

























































' 








^ 
















































A 






























►X 




























































> 






















I< 



























































































^BOEPT, 1 - 



+ 



*' jsigkigLL-i^ 



R1 



fP'|,?"'p"' | ™ 1 



R1 



f 



Figure B.ll -Geometry of the Soil Undeformed Grid Configuration. 



173 



ELEMENT NUMBERS 



14 


14, 


i m: 


1 14 


1 14 


5 146 


147 


143 


149 


150 


13 


n; 


! 13: 


I 13' 


1 13 


5 13S 


137 


133 


139 


140 


12 


12; 


! 12: 


1 12' 


1 12 


5 125 


127 


123 


129 


~130 


11 


ti, 


1 n. 


i 11 


1 11 


5 115 


117 


113 


119 


120 


10 


loi 10: 


1 10 


1 10 


5 iOS 


107 


103 


109 


no 






95 


96 


97 


99 


39 


100 




89 


90 


91 


92 


93 


94 




53 


94 


9S 


35 


57 


99 




77 


73 


79 


90 


91 


92 




71 


72 


73 


74 


75 


7S 




SS 


56 


57 


59 


59 


70 


55 


56 


57 


S3 


59 


SO 


51 


52 ' 


S3 


54 


US 


>45 


47 


49 


49 


50 


SI 


52 * 


S3 


54 


35 


35 


37 


39 


33 


40 


41 


42 


43 


44 


25 


25 


27 


23 


29 


30 


31 


32 


33 


34 


IS 


15 


17 


19 


19 


20 


21 


22 


23 


24 



Figure B.12 - Soil Mesh Elements Number. 



174 



NOCfiL NUM2ESS 



157 


133 


153 


150 


151 


132 


153 


1S4 


153 


136 


145 


u7 . 


,14 3 


143 


ISO 


1S1 


152 


153 


iSU 


155 


1 35 


135 


137 


133 


133 


140 


141 


142 


143 


i44 


«24 


125 


125 


127 


123 


123 


130 


131 


132 


133 


113 


ill 


115 


115 


117 


113 


113 


120 


121 


122 


102 103 1 C4 105 


1CS 


107 


103 


103 


no 


111 




35 


35 


37 


S3 


33 


100 




33 


39 


30 


31 


32 


33 




81 


32 


23 


Sm 


35 


35 




71 175 


75 


77 


73 


73 




57 


S3 


S3 


70 


71 


72 


55 


57 


53 


55 


SO 


61 


52 


53 . 


54 


55 


4S 


h5 


47 


43 


49 


SO 


Si 


S2 


S3 


5n 


34 


35 


33 


37 


33 


33 


40 


41' 


42 


43 


23 


24 


25 


25 


27 


23 


23 


30 


31 


32 


12 


13 


14 


15 


IS 


17 


13 


13 


20 


21 



1 2 3 4 S 5 



10 



11 



Figure B.13 - Soil Mesh Nodal Numbers. 



175 



INCflEn£NT NUtt££R5 



f* 



' / i < t 



////-A 



/ / ,; / / 
/ /'/ / 

t ■ t f *(& 



/ /t / ' 






/ / 1/ / 



// / / / 



/ / / / 
/ y / 



k /// / 



^77? 

/ / / / 



f* 



y / 'i / / 
/ / / / 



/ 






z ' -/ i ' sv ; / * > 



TTi 7 



I—7-7 



iXi 



/ 



/ 



; / 



/ / / 



/ 



/ / / , I / / 



/ 






/ / 



I / 



PT77 



/ / /' / /I/ A / / ,/ 



z 7 ^^ 7 

///•'/ 



7 F7 "~v 






/ /l 



/ I7TJZZZZZZ7 



^77 



/ 



r r 

> / 



T P~7 



/ / / / 1/ / / / 



** 



/ 
/i 



v 7 7 



/* 



n 






/ 



/ 



/ / 
' / 



y / / 



' /V / 



/ / 



/ 



/ 



/ 



A 



/ 
/ / 



/ A 



/ 



/ 



/ 



/ L 



v'/V' 

1 / 



t tit 



/ 



/ / / 



/ / 



■/ / ' / A 



? 7 — 7— 7— TrP 

V ' ' — L 






/ A 
V 
/ 



/ 



/ 



A 



7 



/ 



•/ 



/ 



/ 

/ / 

/ 



7 / T 



' /• 



/ A 



/ / / 



1 r 






/ 



d-/ 



/ 



' / 



/ 

1/ 1 /j 

/ 
/ 



/ 



1 < 



/ 



/ 
/ 



' / ' 



/ 



/ 



/ V / / /- / 



< / / / /« / 



/ 



T 



/ 






/ / / / / 



/i 

/ 



/ 7 



/ 



/ 



/: 



/J—i 



/I 



/■/ 



•m m* rr^ -^r -nT-TTT tn 



' / 1 

' / 
/ / / 
/ / 



7 / 
/ 



/ / 



' / ./ / 



r 7 A 

/// 



/ / 



/ / 



/ / / / / 

/ . / / / / 



A 



w 



A 



^ 



Figure B»1A - Soil Layers Incremental Loading for Trench Installation 
and Mesh Boundary Conditions. 



17b 



APPENDIX C 



SAMPLE OF INPUT DATA AND OUTPUT 



The three sample problems presented here cover the three solution 
levels available when analyzing a reinforced concrete box culvert, that 
is: level 2 box, extended level 2, and level 3. The three samples cor- 
respond to box culverts analyzed during the process of this work. 

Table D.l gives a brief description of the box type, solution level, 
installation type, and some special comments of the sample problems. 
Each problem is presented in the following format: (1) a listing of all 
the input cards, and (2) selected CANDE output for the box responses. 
The soil responses are not presented. 



177 



TABLE C.l Example Problems for Analysis of Reinforced 
Concrete Box Culverts 



Problem 
No. 


Solution 
Level 


Soil 
Installation 


Special Comments 


1 


2 


Embankment 


8*6-8 Box Culvert 
ASTM H = 10 ft. 
stiff linear soil 
soil dead load only 
automatic mesh generation 


2 


2 
Extended 


Embankment 


8*6-8 Box Culvert 

ASTM H = 2 ft. 

stiff linear soil 

soil plus twice HS-20 L.L. 

automatic mesh generation 


3 


3 


N 


6*4-2 Box Culvert 
out-of ground loading 
user's input mesh 



178 



Problem 1 - Input 



CARD TYPE 1...X...10....X...20....X...30....X...40....X...50....X...60....X...70....X...80 



CARD 1A 
CARD IB 
CARD 2B 
CARD 3B-1 
CARD 3B-2 

CARD 2C 
CARD ID, 
CARD 2D 
CARD ID 
CARD 2D 
CARD ID 
CARD 2D 



150.0 


65000.0 


B.O 


8.0 


.01583 


0.50 



ANALYS 2 CONCRE BOX CULVERT 8*6-8 (MEDIUM REINFORCED - H=10 FT) 
-1.0 8.0 STD 3 0.0001 
5000.0 

8.0 8.0 8.0 
.01667 .02417 .02583 
EMBA EMBANKMENT - STIFF SQ-IL 
1 3 9 52.00 

0.0 INSITU-SOIL 
0.33 

0.0 BEDDING-SOIL 
0.33 
120.0 FILL-SOIL 
0.33 



STOP 



1 1 
3333.0 

2 1 
6666.0 

3 1 
3333.0 



40.00 



10.00 



120.00 



12.00 



179 



Problem 1 - Output 

*** PROBLEM NUMBER 1 *«* 
BOX CULVERT. 8*6-8 J MEDIUM REINFORCED - M«10 FT) 

EXECUTION MOUE ANAL 

SOLUTION LEVEL F.C.AUTD 

CULVERT TYPE CONCRETE 



♦♦•NEGATIVE PIPE UIAMETEF IMP. IES NEW CANOE OPTION FOR VARIABLE CONCRETE THICKNESS. *** 
•♦•OPTION IS RESTRICTED TO ANALYSIS CNLY WITH LEVEL 2-BOX, OR LEVEL 3. *** 



PIPE PROPERTIES ARE AS FULL3WS ... 
(UNITS ARE INCH-POUNO SYSTEM ) 

NOMINAL PIPE DIAMETER -1.0000 

CONCRETE COMPRESSIVE STRENGTH 5000.0000 

CONCRETE ELASTIC MODULJS 4266826.00 

CONCRETE PUISSON RATIO 0.1700 

DENSITY OF PIPE IPCF) 150.0003 

STEEL YIELD STRENGTH 65003.0000 

STEEL ELASTIC MODULUS 29000000.0 

STEEL POISSON RATIO 0.3000 

NONLINEAR CUOE (1,2, OR 3 1 3 

CONC. CROCKING STRAIN ( 1.2,3) 0.000100 

CONC. YIELDING STRAIN (2,3) 0.000566 

CONC. CRUSHING STRAIN (2,3) 0.002000 

STEEL YIELDING STRAIN (3) 0.002040 



SPACING LONGITUDINAL REINFORCEMENT 



2.00 



180 



Problem 1 - Output (continued) 



NODE * 


STEEL 


AREAS (IN2) 


STCEL 


CUVERSUN) 


TH1CKNESSUN) 


V 


ASI(N) 


ASO(N) 


THI inj 


TBOIN) 


PTV(N) 


1 


0.0242 


0.0 


1.2500 


1.2500 


8.0303 


2 


0.02*2 


0.0 


1.2500 


1.2500 


3.3000 


3 


0.0242 


0.0167 


1.2500 


1.2500 


8.0000 


4 


0.0242 


0.0167 


1.2 500 


1.2500 


8.3333 


5 


0.0200 


0.0167 


1.2500 


1.2500 


15.0300 


6 


0.0158 


0.0167 


1.2 500 


1.2500 


8.0000 


7 


n.0150 


0.0167 


1.2500 


1.2500 


3.3333 


a 


0.0158 


0.0167 


1.2 500 


1.2500 


8.3000 


<> 


0.0158 


0.0167 


1.2 500 


1.2500 


8.3033 


13 


0.0158 


0.0167 


1.2 500 


1.2500 


8.0000 


11 


0.0208 


0.0167 


1.2 500 


1.2500 


16.0000 


12 


0.0258 


0.0167 


1.2 500 


1.2500 


8.0033 


13 


O.0258 


0.0167 


1.2 500 


1.2500 


8.0000 


14 


0.0258 


0.0 


1.2500 


1.2500 


fl.0000 


15 


0.0253 


0.0 


1.2500 


1.2500 


8.0000 



* * BEGIN GENERATION UF CANNED MESH * * 

THE DATA TO BE RUN IS ENTITLED 

EMBANKMENT - STIFF SOIL 

TYPE UF MESH EMBANKMENT 

PLOTTING DATA SAVED 

PRINT SOIL RESPONSES 1 

PRINT CONTROL FOR PREP OUTPUT 3 

NUMBER OF CONSTRUCTION INCREMENTS <J 

SPAN Or BOX 104.00 

HEIGHT OF BOX 80.00 

SOIL ABOVE TOP OF BOX (FT) 10.00 

*1ESH HEIGHT ABOVE TOP CF BOX IFT) 10.00 

SOIL DENSITY ABOVE MESH (PCF) 120.00 

IDENTIFICATION OF MATERIAL ZONE WITH MATERIAL NUMBER 

MATERIAL-ZONE MATERIAL NO. 

INS ITU 1 

BEDDING 2 

FILL 3 



181 



Problem 1 - Output (continued) 

* * BEGIN PREP OF FINITE ELEMENT INPUT * * 

THE DATA TO BE RUN IS ENTITLED 
EMBANKMENT - STIFF SOIL 

NUMBER OF CONSTRUCT UN INCREMENTS 9 

PRINT CONTROL FOR PREP OUTPUT 3 

INPUT DATA CHECK 

PLOT* TAPE GENERATION 

ENTIRE FINITE ELEMENT RESULTS UUTPUT 1 

THE NUMBER OF NODES IS 167 

THE NUMBER OF ELEMENTS IS 150 

THE NUMBER UF BOUNCARY CUNDITIUNS IS 200 



MATERIAL CHARACTERIZATION F3* SOILS. 

PROPERTIES FOR MATERIAL I ******** INSITU-SUIL 

DENSITY = 0.0 

YOUNGS MODULUS" 0.3333E+0* 
POISSUNS RATIO= 0.3300C+00 
CONFINED MOD.= 0.4938E+0^ 
LATERAL COEFF.= 0.<t925fc+00 

PROPERTIES FOR MATERIAL 2 ******** OEUOING-SOIL 

DENSITY = 0.0 

YOUNGS MODULUS* 0.6666E+04 
POISSONS RAT 10= 0.3300E+00 
CUNFINEU MUD.= 0.9877E+04 
LATCRAL CUEFF.= O.^925E+0O 

PROPERTIES FOR MATERIAL 3 ******** FILL-SOIL 
DENSITY = 0.12000E+03 

• 

YOUMGS MODULUS* 0.3333E+0* 
POISSONS RATION 0.3300E+00 
CONFINED MOD.= 0.4938E+04 
LATERAL COEFF.= 0.4925E+00 



182 



Problem 1 - Output (continued) 



STRUCTURAL RESPONSE OF CULVERT FOR LOAD INCREMENT 



COORDINATES. DISPLACEMENTS AND CRACK DEPTHS ARE IN INCHES 
PRESSURES ARE IN LB/IN**2 
MOMENTS ARE IN IN.*LB/IN. 
THRUST AND SHEAR ARE IN LO/IN. 



NPPT 



10 



11 



X-COORD. 
Y-COURD. 


X-OISP. 
Y-OISP. 


N-PRES. 
S-PRES. 


MOMENT 
THRUST 


SHEAR 
CRACK DEPTH 


0.0 
40.00 


0.0 
-O.36327E+00 


-0.82420E+01 
0.0 


0.55599E*04 
-0.11517E+03 


0.0 
0.60566E+01 


13.00 
40.00 


-0.48848E-04 
-0.36062fcf00 


-0.85117E+01 
-0.16894E+00 


0.48633E+04 
-0.11626E+03 


0. 10890E+03 
0.0 


26.00 
40.00 


-0.9081 7E-04 
-0.35366E+00 


-0.91547E+01 
-0.37016E+00 


0.27284E*04 
-0.11977E+03 


0.22374E+03 
0.0 


39.00 
40.00 


-0.13419E-03 
-0.34457E+00 


-0.10687E+02 
-0.123UE+00 


-0.948B0E+03 
-0. 12297E+03 


0.35271E*33 

0.0 


52.00 
40.00 


-0.16387E-03 
-0.33563E+00 


-0.S0348E+01 
-0 . 10693E + 02 


-0.64476E+04 
-0.36492E+03 


0.14262E+03 
0.0 


52.00 
26.67 


0.79563E-02 
-0.33548E+00 


-0.44441E+01 
-0.45603E+01 


-0.46375E+04 
-0.63 646E+03 


-0.10731E+03 
0.0 


52. OC 
13.33 


0.13600E-01 
-0.33524E+00 


-0.50773E+01 
-0. 39506 E+Ol 


-0.36019E«-04 
-0.69321E+03 


-0.43829E+02 
0.0 


52.00 
0.0 


0.15840E-01 
-0.33498E+00 


-0.62997E+01 
-0.28920E+01 


-0.34689E+04 
-0.73882E+03 


0.32017E+02 
0.0 


52.00 
-13.33 


0.14765E-01 
-0.33470E+00 


-0.65201E+OI 
-0.21085F+01 


-0.44556E+04 
-0.77216E+03 


0.11748E+03 
0.0 


52.00 
-26.67 


0.91028E-02 
-0.33438E*00 


-0. 70390E+01 
-0.84638E+00 


-0.66014E*04 
-0.79186E+03 


0.20788E«-03 
0.60218E + 01 


52.00 
-40.00 


-0. 44758 E-03 
-0.3341 7E+00 


-0. 15362E + 02 
0.33181E+01 


-0. 10001E+05 
-0.56033E+03 


-0. 16154E4-03 
0.0 



t2 



39.00 -0.27012E-03 -0.14983E + 02 -0.24900E+04 -0.4804 9E«-03 
•40.00 ~0.32325E*03 0.5577lt*00 -0.31954E*03 0.0 



13 



26.03 -0.25810E-03 -0.12640E+02 0.24882E+04 -0.30094E+03 

■40.00 -0.31149E+00 0.94244E»00 -0.30979E+03 0.0 



14 



13.00 -0.14314E-03 -0.11427E+02 0.53298E+04 -0.14451E+03 

■40.00 -0.30161E*00 0.53084E+00 -0.30021E+03 0.54325E+01 



15 



0.0 0.0 -0.10805E+02 0.624L1E+04 0.0 

•40.00 -0.29753E+00 3.0 -0.29676E+03 0.58268E+01 



183 



Problem 1 - Output (continued) 



STRESSES IN CULVERT WALL (PS I ) FCR LOAD INCREMENT 





ELLIP. OP 








PT 


INNER CAGE 


OUTER CAGE 


CONCRETE 


SHEAR 




STEEL 


STEEL 


COMPRESSION 


STRESS 


1 


O.34614E+05 


0.0 


-0. 13187E+04 


0.0 


2 


0.20485E+04 


0.0 


-0 .46337E+03 


0. 16134E+02 


3 


0. 13956E+04 


-0.13196E+04 


-0.25882E+03 


0.33146E+02 


4 


-0.52378E+03 


0.31524E+03 


-0.9B963E+02 


0.52253E*-02 


5 


-0. 10500E+04 


0.72864E+03 


-0. 16826E*03 


0.96693E«-01 


6 


-0.26324E+04 


0. 15131E+04 


-0. 49515E+03 


-0. 15897E*02 


7 


-0.22193E+04 


0.10012E+04 


-0.40881E+03 


-0.6493lE*0l 


9 


-0.21999E+04 


0.90183E+03 


-0.40239E+03 


0.47433E+01 


9 


-0.26 704E+04 


0.13131E+04 


-0 .49531E+03 


0.17405E+02 


10 


-0.26859E+04 


0.41673E+05 


-0. 17685E*r»4 


0.30797E+02 


11 


-0.16246E+ K 


0. 11323E+04 


-0.26039E+03 


-0. 10952E+02 


12 


-0. 13679C+04 


0.82989E +03 


-0.25867E+03 


-0.71184E+02 


13 


0.82146E+03 


-0. 13772E+04 


-0.25998E+03 


-0.44584E*02 


I* 


0.23769E+05 


0.0 


-0. 13087E+04 


-0.21408E+02 


15 


0.3234 8E+05 


0.0 


-0. 14766E* 04 


0.0 



STRAINS IN THE INNER ANU OUTER FIBER OF THE CULVERT WALL 
(ONLY STRAINS FOR COMPRESSION ZLNES HAVE PHYSICAL MEANING) 



NPPT 



INNER STRAIN 



OUTER STRAIN 



I 


0.13426E-02 


-0.29872 E-03 


2 


0.95622E-04 


-0. 10497E-03 


3 


1. 5 1 603 E- 14 


-0.58631E-04 


4 


-0.22419E-34 


0.15875E-C4 


5 


-0.3811 7E-04 


0.28032E-04 


6 


-0. 11217E-03 


0.77045E-04 


7 


-0.92608E-04 


0.54384E-04 


8 


-0.9H53E-04 


0.5042 JE-04 


9 


-0.1122 0E-03 


0.69614E-0't 


10 


-0.'»0063E-D3 


0.16240E-02 


11 


-0.58987E-04 


0.43540E-04. 


12 


-0.58597E-04 


0.41715E-04 


13 


0.41457E-04 


-0.58895E-04 


14 


0.93886E-D3 


-0.29646E-03 


15 


0.12650E-02 


-0.33450E-03 



184 



Problem 1 - Output (continued) 



CALCULATED SAFETY FACTORS FUR LOAD INCREMENT 9 

STEEL YIELD STRESS / MAX. STEEL STRESS 1.560 

CONCRETE STRENGTH / MAX. COMPRESSIVE STRESS .... 2.827 

WALL SHEAR CAPACITY / MAX. SHEAR 1.987 

PERFORMANCE FACTORS 

0.C1 INCH / MAX. CRACK WIDTH 1.215 



+ * * * NORMAL EXIT FRUM CANDE * * * * 



185 



Problem 2 - Input 



CARD 


TYPE 


1 














CARD 


1A 


ANALYS 2 CONCRE BOX CULVERT 3*6-3 


- HS-20 


LIVE LOAD 


(MINIMUM 


CARD 


IB 




-1.0 


8.0 STD 3 


0.0001 








CARD 


223 




5000.0 




150.0 


6500 




CARD 


3B-1 




8.0 


8.0 8.0 


8.0 




8.0 




CARD 


33-2 




.02583 


.03833 .02917 


.01583 




0.65 




CARD 


1C 


EMBA EMBANKMENT - STIFF SOIL 










CARD 


2C 




1 


3 8 52.00 


40.00 




2.00 


120.00 


CARD 


4C 






2 










CARD 


7C 




124 


-60 


.0 




7 




CARD 


7C 




124 


-51 


.1 




8 




CARD 


ID 




1 1 


0.0 INSITU-SOIL 










CARD 


2D 




3333.0 


0.33 










CARD 


ID 




2 1 


0.0 BEDDING-SOIL 










CARD 


2D 




6666.0 


0.33 










CARD 


ID 


L 


3 1 


120.0 FILL-SOIL 










CARD 


2D 




3333.0 


0.33 











.70....X...80 



MOD 

12.00 



STOP 



186 



, 



Problem 2 - Output 

• •• PROBLEM NUMBER 1 **♦ 

BOX CULVERT 8»6-8 - HS-20 LIVE LOAD (MINIMUM SUIL CUVERI 

EXECUTION MODE ANAL 

SOLUTION LEVEL F.E.AUTO 

CULVERT TYPE CUNCRETE 

••♦NEGATIVE PIPE OIAMETER IMPLIES NCW CANOE OPTION FOR VARIABLE CONCRETE THICKNESS. *•* 
•♦•OPTION IS RESTRICTED TO ANALYSIS CNLY WITH LEVEL 2-BUX, OR LEVEL 3. *** 



PIPE PROPERTIES ARE AS FOLLOWS ... 
(UNITS ARE INCH-POUND SYSTEM » 

NOMINAL PIPE OIAMETER -I. OOOO 

CONCRETE COMPRESSIVE STRENGTH 5000.0000 

CONCRETE ELASTIC MUDULJS 4286826. 00 

CONCRETE PUISSUN RATIU 0. WOO 

DENSITY OF PIPE (PCF) 150.0000 

STEEL YIELO STRENGTH 65000.0000 

STEEL ELASTIC MODULUS 29000000.0 

STEEL POISSUN RATIO 0.3000 

NONLINEAR COOE (1,2, OR 3) 3 

CONC. CRACKING STRAIN (1,2,3) 0.000100 

CONC. YIELDING STRAIN (2,31 0.000566 

CONC. CRUSHING STRAIN 12,3) 0.002000 

STEEL YIELOING STRAIN (3) 0.002040 



SPACING LONGITUDINAL RE I NFCPCEMENT 



2.00 



187 



Problem 2 - Output (continued) 



13DE * 


STEEL 


AREASUN2) 


STEEL 


COVERSUNI 


THICKNESS! 


N 


ASIIN) 


ASO(N) 


TL> I ( N ) 


TMMN.I 


PTV(N) 


1 


0.0383 


0.0 


1.2500 


1.2500 


8.0000 


2 


0.0383 


0.0 


1.2500 


1.2500 


8.00 33 


3 


0.03 A3 


0.0258 


1.2 500 


1.2503 


8.0333 


4 


0.0333 


0.0258 


1.2 500 


1.2500 


8.0303 


5 


0.0271 


0.0258 


1.2500 


1.2500 


1ft. 0000 


6 


0.0158 


0.0258 


1.2500 


1.2500 


9.3303 


7 


0.0158 


0.0258 


1.2 500 


1.2 500 


8.0C00 


8 


0.0158 


0.0258 


1.2500 


1.2500 


8.0000 


9 


0.0158 


0.0258 


1.2500 


1.2 500 


8.0000 


10 


0.0158 


0.0258 


1.2 500 


1.2500 


fl.OCOO 


11 


0.0225 


0.0258 


t.2 500 


1.2500 


16.0033 


12 


0.0292 


0.0258 


1.2500 


1.2500 


8.0003 


13 


0.0292 


0.3258 


1.2 500 


1.2 500 


8.0000 


14 


0.0292 


0.0 


1.2500 


1.2503 


8.0333 


15 


0.0292 


0.0 


1.2500 


1.2500 


8.0000 



• * BEGIN GENERATION OF CANNEC MESH * * 

THE DATA TO BE RUN IS ENTITLED 

EMBANKMENT - STIFF SOIL 

TYPE 3F MESH EMBANKMENT 

PLOTT ING DATA SAVED 

PRINT SOIL RESPONSES I 

PRINT CONTROL FOR PREP OUTPUT 3 

NUMBER OF CONSTRUCTION INCREMENTS 8 

SPAN OF BOX 134.00 

HEIGHT OF BOX 80.00 

SOIL ABOVE TOP OF BOX (FT) 2.(0 

^ESH HEIGHT ABOVE TOP CF BOX (FT) 2.00 

SOIL DENSITY ABOVE MESH IPCF1 120.00 

IDENTIFICATION OF MATERIAL ZONE WITH MATERIAL NUMBER 
MATERIAL-ZONE MATERIAL NO. 

INS ITU 1 

BEDDING 2 

FILL 3 



188 



Problem 2 - Output (continued) 



* * BEGIN PREP OF FINITE ELEMENT INPUT * * 

THE DATA TO BE RUN IS ENTITLED 
EMBANKMENT - STIFF SOIL 

NUMBER OF CONSTRUCTION INCREMENTS 

PRIMT CONTRCL FOR PPEP OUTPUT 3 

INPUT DATA CHECK 

PLOT TAPE GENERATION 

ENTIRE FINITE ELEMENT RESULTS OUTPUT I 

THE NJMEER OF NOOES 13 13* 

THE NUMBER OF ELEMENTS IS 120 

THE MUMBER OF BOUNCARY CONDITIONS IS 200 



* * * CHANGES TO STANCARO LcVEL 2 MfcSH * * * * 

* NUMBER OF NOOES TO BE CHANGED 0* 

* NUMBER OF ELEMENTS TU BE CHANGED 0* 

* ADDITIONAL BOUNOARY CGNUITIONS 2* 

******************* **•* 



♦ ♦•ADDITIONAL BOJNCARY COND IT IONS . . .P 3 R CES = LHS t DISPLACEMENTS = INCHES.. 



BOUNOARY 


LOAD 


X-FORCE 0* 


Y-FORCE OR 


X-Y PU TAT I UN 


NOOE 


STEP 


X-OISPLACEMENI 


Y-DISPLACEMENT 


UEGFEES 


12* 


7 


F a 0.0 


F = -0.6000C*02 


)."> 


12* 


8 


F » 0.0 


F = -0.5U0E+02 


0.0 



189 



Problem 2 - Output (continued) 



MATERIAL CHARACTER IZAT 1UN FOR SUILS. 

PROPERTIES FOR MATERIAL 1 ♦*«»**♦* INSITU-SLUL 

DENSITY = 0.0 

YOUNGS MO0ULUS= 0.3333EKK 
POISSONS RAT 10= 0.3300C*00 
CONFINED MOl).= 0.4938E*04 
LATERAL COEFF.= U.4925E+00 

PROPERTIES FOR MATERIAL 2 *♦*♦*»♦♦ BELUING-S01L 

DENSITY = 0.0 

YOUNCS MODULUS" 0.6666E*04 
POISSONS RATIO= 0.3300E*00 
CONF INEO MOO." 0.9877E+04 
LATERAL COEFF.= 0.49251*00 

PROPERTIES FOR MATERIAL 3 *♦*••*♦♦ TILL-SOIL 

DENSITY = 0. 12U00E*33 

YOUNGS MODULUS" 0.3333CKK 
POISSONS PATIO= 0.3300E*00 
CONF INED MOO." 0.4938E*04 
LATERAL COEFF." 0.<t925l*00 



190 



Problem 2 - Output (continued) 



STRUCTURAL RESPONSE Of CULVERT FOR LOAD INCREMENT 8 

COORDINATES. 01 SPLACEMENTS AND CRACK DEPTHS ARE IN INCHES 
PRESSURES ARE IN LB/IN**2 
MOMENTS ARE IN IN.*LB/IN. 
THRUST ANO SHEAR ARE IN LD/IN. 

NPPT X-COORO. X-OISP. N-PRES. MUMENT SHEAR 

Y-COORO. Y-DISP. S-PRES. THRUST CRACK DEPTH 

1 0.0 0.0 -0.82803C*0l J. 40991 fc *0'. J. 3 
40.00 -0.1930<JE*00 0.0 0.25164E*02 0.0 

2 13.00 0.89571E-G5 -0 . I 5124E* 01 0.33993E+04 0.96151C*02 
40.00 -0.19143E*00 -0.17414E*01 0. 13845E*02 0.0 



13 



11 



13 



14 



15 



26.00 


0.98505C-G5 


-0.34548E*0l 


0. 159«HE *04 


0. 1609'»E*03 


40.00 


-0. 18703E*00 


-3.1<>975E*01 


-0. 104?JE*02 


0.3 


3<».00 


0.16997E-05 


-0.23285E*0l 


-0.78224E+03 


0.19853E + 03 


40.00 


-0.1«138E*00 


-3.42829E*r»0 


-3.26?25C*32 


3.0 


52.00 


-0.51747E-05 


-0. 16996E*01 


-0.35617EO4 


0.85171E*32 


40.00 


-0.17587E*00 


-3 ,33O03E*Ol 


-0.144 70L : *03 


3.0 


52.00 


0.50454C-02 


-0.9 7901C*00 


-0.29°05E*0't 


-O.36796E+02 


26.67 


-0.17581 fc*00 


-0.20297E*01 


-0.27393E+03 


0.3 


52.00 


0.84971E-02 


-0. 113 26E*01 


-0.258711*04 


-0.22713C*02 


13.33 


-0. 17571E+00 


-3.23598C+01 


-0.30319E+02 


0.0 


52.00 


0.95969E-02 


-0.22470E*Ol 


-0.2 3*4OE*04 


-0. 18745E*C0 


0.0 


-0.1 7559E*CJ 


-0.20561O01 


-3.33263L03 


0. J 


52.00 


0.84966E-02 


-0.23406C* r >l 


-0.25H21E H4 


3.3 )?V/F*J? 


13.33 


-0. 17546E*00 


-0.2056^C*Ol 


-0.36005EO3 


0.0 


52.50 


0.531 64C-C2 


-3. 317131*11 


-o. »i95it*04 


O.67l4in»07 


26.67 


-0.17533E*00 


-0. 155541: ♦Ol 


-0.3 0413C*03 


0.0 


52.00 


-0.16512C-03 


-0.792 491* 11 


-0.<.3 75">C»0'» 


-■1.101250*03 


40.00 


-O.I /523E*00 


3 .2275'.C»ni 


-0.26?fc9t*93 


0.0 


26.00 


-0.079 13E-04 


-0.627I6E*Ol 


3.19106E*04 


-3.15253E+03 


40.00 


-0.I6368E*00 


0.^.0764E*no 


-0.12 H9L'*03 


0.0 


13.00 


-0.43730E-04 


-0.57943E+O1 


0.33603E+04 


-0.74104E*02 


•40.00 


-0.15934E+00 


0.20350E*00 


-0.123Z2E*03 


0.0 , 


O'.O 


0.0 


-0.56062E+01 


0.383Z2C*04 


0.3 


•40.00 


-0.15774E*00 


3.0 


-0.12190E*03 


3.3 



191 



Problem 2 - Output (continued) 



STRESSES IN CULVERT WALL (PS I ) FOR LUAO INCREMENT 



NPPT 



1 

2 

3 

'♦ 

5 

6 

7 

8 

9 

10 

11 

12 

13 

1* 

15 



ELL1P. OR 

INNER CAGE 
STEEL 

0.17e47E*04 

0. I 474 1E*04 

3.67317E03 

-0.35505E*03 

-C.54772E+03 

-0.15723E + 04 

-0. 14181E*04 

-0.1353 7E04 

-0. 14655E+C4 

-0.17599E+04 

-3.71466E*03 

-0.37275E+03 

0.72027E*03 

0. 13637E04 

0.15715E*04 



OUTER CAGE 
STLbL 

0.0 
0.0 

-O.7O528C*03 
0.31755E*03 
0.42065E »03 
0. 10729C+04 
0.870BOE+03 
0.75670E + 03 
0.81929E*03 
0.10670E+04 
0.47993E«-03 
0.1482 3E*03 

-0.943 74£*03 
0.0 
0.0 



CONCRETE 
;OMPRESSION 

■0. 37247E+03 
•0. 30974E + 03 
■0. 14109E*03 
■0.70357E* 02 
-3.8829lE*02 
-0.30107F>03 
-0.26849E*03 
-0.2 53 < ;5E»-O3 
O .27494E*03 
-0.33277E* 03 
-3. U432E*0j 
-0.68036E*02 
-3. 1831 IF*03 
-0.32448E*OJ 
-0 .36778E* 03 



SMEAR 
STRESS 

0.0 

0. 14245E+02 

0.23843E*02 

0.2 Wl2fc*02 

0.57743E*0I 

-0.54512E«-01 

-0.3365fcb»01 

-J.27771G-01 

0.450321*01 

0.99471F.»0l 

-0.68642E+01 

-0. 35853L«-02 

-0.22597C»02 

-0.1 197BC»02 

0.0 



STRAINS IN THE INNER AND OUTER FIBER OF THE CULVERT WALL 
(ONLY STRAINS FOR COMPRESSION ZCNES HAVE PHYSICAL MEANING) 



NPPT 



INNER STRAIN 



OUTER STRAIN 



I 


0.8 t9<J9E-0 4 


-0.843 75E- 


■C4 


2 


0.6781 5E-34 


-0.70166E- 


■04 


3 


0. 30954E-04 


-0.31962E- 


■04 


4 


-0.15938E-04 


0.14761T- 


■C4 


5 


-0.20001E-34 


0. 16013E- 


■04 


6 


-3.68232E-34 


0.52530E- 


■04 


7 


-0.60822E-04 


0.43649C- 


■04 


8 


-0.57529E-04 


0.38795E- 


-04 


9 


-O.A2283E-04 


0.420U4E- 


■04 


10 


-0.75384E-04 


0.53641C- 


-C4 


11 


-0.25897E-14 


3. 10531b- 


■04 


12 


-0. 15412E-04 


0.83669E- 


-C5 


13 


0.34469E-04 


-0.41481 E- 


■04 


14 


3.64327E-34 


-0. 73535 E- 


•04 


15 


0.73873E-04 


-0.83314E- 


■04 



192 



Problem 2 - Output (continued) 



CALCULATED SAFETY FACTORS FOR LOAD INCREMtNT ft 

STEEL YIELO STRESS / MAX. STEEL STRESS 36.*21 

CCNCRETE STRENGTH / MAX. COMPRESSIVE STFESS .... 13. *24 

WALL SHEAR CAPACITY / MAX. SHEAR 3.9** 

PERFORMANCE FACTORS 

0.01 INCH / MAX. CRACK WIDTH 9909.996 



* • * * NORMAL EXIT FROM CA>iDE * • * * 



193 



Problem 3 - Input 



CARD 


TYPE 


















CARD 


1A 


ANALYS 


3 CONCRE BOX CULVERT 


TEST 


(6*4-2) 


TEST-1 


CARD 


IB 




-1.0 


7 


.0 


ARBZ 


3 


0.0001 




CARD 


2B 


6965.0 









.17 


150.0 


99430.0 


CARD 


3B 


.03475 


.00000 


1.' 


443 




7.375 


CARD 


3B 


.03475 


.03550 


1.' 


443 




7.375 


CARD 


3B 


.03475 


.03550 


1. 


443 




7.375 


CARD 


3B 


.03475 


.03550 


1.443 




7.375 


CARD 


3B 


.01737 


.03550 








14.0 


CARD 


3B 


.00000 


.035 


50 










CARD 


3B 


.00000 


.03550 










CARD 


3B 


.00000 


.035 


50 










CARD 


3B 


.00000 


.03550 










CARD 


3B 


.00000 


.035 


50 










CARD 


3B 


.01337 


.03550 








14.0 


CARD 


3B 


.o: 


2675 


.035 


50 


1.006 




7.438 


CARD 


3B 


.02675 


.035 


50 


1.006 




7.438 


CARD 


3B 


.0. 


2675 


.03550 


1.006 




7.438 


CARD 


3B 


.0. 


2675 


.00000 


1.006 




7.438 


CARD 


1C 


PREP 


BOX CULVERT 


OUT-OF-GROUND STUDY 


(TEST 6*- 


CARD 


2C 


10 


3 








1 


17 


IS 14 




CARD 


3C 


1 





0.0 




27.5 








CARD 


3C 







9.0 




27.5 








CARD 


3C 


3 





19.0 




27.5 








CARD 


3C 


4. 





29.0 




27.5 








CARD 


3C 


5 





39.5 




27.5 








CARD 


3C 


6 


a 


39.5 




17.0 








CARD 


3C 


7 





39.5 




8.5 








CARD 


3C 


8 





39.5 




0.0 








CARD 


3C 


9 





39.5 




-8.5 








CARD 


3C 


10 





39.5 




-17.0 








CARD 


3C 


11 





39.5 




-27.5 








CARD 


3C 


12 





29.0 




-27.5 








CARD 


3C 


13 





19.0 




-27.5 








CARD 


3C 


14 





9.0 




-27.5 








CARD 


3C 


15 





0.0 




-27.5 








CARD 


3C 


16 





6.0 




-33.5 








CARD 


3C 


L 17 





12.0 




-33.5 








CARD 


4C 


1 


2 


1 








1 






CARD 


4C 


2 


3 


2 








2 






CARD 


4C 


3 


4 


3 





. o 


3 






CARD 


4C 


4 


5 


4 








4 






CARD 


4C 


5 


6 


5 








5 






CARD 


4C 


6 


7 


6 








6 






CARD 


4C 


7 


8 


7 








7 






CARD 


4C 


8 


.9 


8 








8 






CARD 


4C 


9 


10 


9 








9 






CARD 


4C 


10 


11 


10 








10 






CARD 


4C 


11 


12 


11 








11 






CARD 


4C 


12- 


13 


12 • 








12 






CARD 


4C 


13 


14 


13 








13 






CARD 


4C 


14 


15 


14 








14 


■ 




CARD 


4C 


L IS 


16 


17 


14 





1 







.60....X...70....X...80 
1415 



194 



Problem 3 - Input (continued) 

CARD TYPE 1.. .X...10....X...20....X...30....X...40. ...X...50....X...60.. ..X...70....X...80 

CARD 5C 11 0.0 1 

CARD SC IS 1 0.0 1 

CARD 5C 16 1 0.0 1 

CARD 5C 17 1 0.0 1 

CARD SC 2 -100.0 1 

CARD SC 2 -100.0 2 

CARD SC 2 -100.0 3 



0.0 






0.0 








1 


0.0 




1 


0.0 







-100.0 







-100.0 







-100.0 







-100.0 







-100.0 







-100.0 







-100.0 







-100.0 







-100.0 







-100.0 



CARD SC 2 -100.0 4 

CARD SC 2 -100.0 3 

CARD SC 2 -100.0 6 

CARD 5C 2 -100.0 7 

CARD SC 2 -100.0 8 

CARD SC 2 -100.0 9 

CARD SC L 2 -100.0 10 

CARD ID L 1 1 ' 

CARD 2D 1.0E+11 0.0 
STOP 



195 



Problem 3 - Output 

♦ ♦• PROBLEM NUMBER i *•* 

BOX CULVERT TEST «6**-2J TEST-l 

EXECUTION MUOE ANAL 

SOLUTION LEVEL F.E.USE* 

CULVERT TYPE CONCRETE 

♦♦♦NEGATIVE PIPE DIAMETER IMPLIES NEW CANOC OPTION FOR VARIABLE CONCRETE THlCKNESi. ♦ ♦* 
♦•♦OPTION IS RESTRICTED TO ANALYSIS CNLV WITH LEVEL 2-BOX, OR LEVCL 3. ♦♦♦ 



PIPE PROPERTIES ARE AS FOLLOWS ... 
(UNITS ARE INCH-POUND SYSTEM ,) 

NOMINAL PIPE DIAMETER -I. OOOO 

CONCRETE COMPRESSIVE SMENGTh 6965. OCOO 

CONCRETE ELASTIC MUOULUS 505*5*4.00 

CONCRETE POISSON RATIO 0.1700 

DENSITY OF PIPE (PCF) 150.0000 

STEEL YIELD STRENGTH 99A30. 0000 

STEEL ELASTIC MODULUS ^ 29000000.0 

STEEL PCiSSUN RATIO 0.3000 

NCNLINEAR CODE (1,2, OR 3) 3 

CONC. CRACKING STRAIN (1,2.3) 0.000130 

CONC. YIELDING STRAIN (2,31 0.000668 

CONC. CRUSHING STRAIN (2,3) 0.002000 

STFEL YIELOING STRAIN (3) 0.003120 



SPACING. LONGITUDINAL RE I NFORCEMENT 



2.00 



196 



Problem 3 - Output (continued) 



njoe * 



STEEL AREASIIN2) 



STEEL COVERS! IN) 



THICKNESS! IN » 



N 


ASIfNI 


ASO(N) 


TBI IN) 


TDCXN) 


PTV(N) 


I 


0.0347 


0.0 


1.4430 


1.2500 


7.3750 


2 


0.0347 


0.0355 


1.4430 


1.2500 


7.3750 


3 


0.0347 


0.0355 


1.4430 


1.2500 


7.3750 


4 


0.0347 


0.0355 


1.44J0 


1.2500 


7.3750 


5 


0.0174 


0.0355 


1.2 500 


I. 2503 


14. JO-JO 


6 


0.0 


0.0355 


1.2 500 


1.2500 


7. 1003 


7 


0.0 


0.0355 


1.2 500 


1.2500 


7.0000 


9 


0.0 


0.0355 


1.2500 


1.2500 


7.0000 


n 


0.0 


0.0355 


1.2500 


1.2500 


7.0000 


10 


0.0 


0.0355 


1.2500 


1.2500 


7.0000 


11 


0.0134 


0.0355 


1.2500 


1.2500 


14.0003 


12 


0.0267 


0.0355 


I .0060 


1.2500 


7.4380 


13 


0.0267 


0.0355 


1.0060 


1.2500 


7.4380 


14 


0.3267 


0.0355 


1.0 ObO 


1.2500 


7.4380 


15 


0.0267 


0.0 


1.0060 


1.2500 


7.4380 



* * 8EGIN PREP OF FINITE ELEMENT INPUT * * 

THE OATA TO. OE RUN IS ENTITLED 

BOX CULVERT CUT-UF-GRUUNU STUOY I TEST 6*4-2) 

NUMBER OF CONSTRUCTION INCREMENTS 10 

PRINT CONTROL FOR PREP OUTPUT 3 

INPUT OATA CHECK 

PLUT TAPE GENEFATION 

ENTIRE FINITE ELEMENT RESULTS OUTPUT — - 1 

THE NUMBER OF NOOES IS 17 

THE NUMBER OF ELEMENTS IS 15 

THE NUMBER OF BOUNCARY CUNDITIUNS IS 14 



197 



Problem 3 - Output (continued) 



♦ ••BOUNDARY CONDI T IONS .. .FORCES * LBS DISPLACEMENT S » INCHES... 



8OUN0ARY 
NODE 



1 

15 

16 

17 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 



LOAO 


X-FORCE 


0* 




Y- 


FORCE OR 


STEP 


X-OISPLACEMEMT 




Y- 


OlSPLACtMENT 


I 


D » 0.0 




F 


3 


0.0 


1 


» 0.0 




F 


S 


0.0 


1 


F « 0.0 







■S 


0.0 


1 


F • 0.0 







= 


0.0 


I 


F » 0.0 




F 


= 


-O.lOuOE«-03 


2 


F » 0.0 




F 


a 


-0.1000L+03 


3 


F * 0.0 




F 


s 


-0.1000C+03 


4 


F » 0.0 




F 


a 


-O.IOOOE*03 


5 


F » 0.0 




F 


■ 


-0.1000C*03 


6 


F = 0.0 




F 


= 


-O.IOOOE*03 


7 


F ■ 0.0 




F 


a 


-0.1000E+03 


8 


F » 0.0 




F 


= 


-0.1000C*03 


9 


F a 0.0 




F 


3 


-0.1000E+03 


10 


F * J.O 




F 


3 


-0.1000E*03 



X-Y ROTATION 
OE^EES 



0.0 

0.) 

0.0 
0.0 
O.J 
0.0 
0.0 
0.0 
0.0 
0.5 
0.0 
0.1 
0.1 
O.rt 



198 



Problem 3 - Output (continued) 



STRUCTURAL RESPONSE UF CULVERT FOR LOAO INCREMENT 5 

COORDINATES t DISPLACEMENTS AND CRACK DEPTHS ARE IN INCHES 
PRESSURES ARE IN LB/IN**2 
MOMENTS ARE IN IN.*LB/IN. 

TH<US7 AND SHEAR ARE IN LB/IN. 

NPPT X-COORD. X-OISP. N-PRFS. MOMENT SHEAR 

Y-COORO. r-DISP. S-PRES. THRUST C«ACK DEPTH 

1 0.0 0.0 -0.6441CE*<>0 0.91441C*04 0.0 

27.50 -0.15773E*00 3.0 0.l3239E«-02 0. 56336E«-0l 

2 9.00 0.21162E-05 -3.«3278E»02 0.9ll7flE«-0* 0.25597E*03 
27.50 -3.l4942t*00 -3.18370E-05 3.13239E*02 3.57301EOI 

3 19.00 0.50681E-05 -0.65051F*00 0.4027OEKK 0.51229EO3 
27.50 -0.12B14E*00 0.17166E-05 0.13239E*02 0.0 

4 29.00 0.83627E-05 -3.62088C*00 - J. 1 1200E*0« 0.51872E*03 
27.50 -0.10406E+00 0.26052C-05 0.13239E*02 0.0 

5 39.50 0.10792E-G4 -0.46690t*00 -0.66008E*04 3.2675<>C»03 
27.50 -0.78849E-01 -0 .461 1 8E*00 -0.25778E*03 0.0 

6 39.50 0.24313E-01 0.34437E-02 -0.67'»93E*0<» 0.12265E*02 
17.00 -0.78740E-01 -3.63980E»03 -0.53184E*03 0.47313COI 

7 39.50 0.39382E-01 0.29297E-02 -0.6B619E»04 0. 1 32 36C 02 

8.50 -0.785*6E-0l -0.64936E»no -0.53763EO3 0. 474J5E/01 

8 39.50 0.45814E-01 3.51970O03 -3.69743E*r>'. 0.13221FO2 

0.0 -0.78341E-01 -0.6452?E*00 -0. 5'.3 1 4E *03 0.47867CO1 

9 39.50 0.4200JE-01 0.378071-02 -0. 7006 7 E »0<. 0.13231T»02 
-8.50 -0.7bl09E-01 -0 . 6<.568E»00 -0. 5<.P62t »02 0. «. 7<)<,2C ♦ 01 

10 39.50 0.26661E-01 -0.7863VE-O2 -0. 71 90 9L *04 0.1322'»E*02 
-17.00 -0.77876fc-0l -0 .64 LOH^ J -0 . 55'.'. I E 03 J.'.OOl '»C *0 1 

11 39.50 -0.9O*92L-O5 . 45"06F *00 -0.73 3801*0'. -0. ? 75 sor »03 
-27.50 -0.77757C-01 -3.45692t*0J -0. 205 3 5C»03 n.o 

12 29.00 -0.A619JE-05 0.6M27r«-O3 - 3 . 1 '. I 3 I L ♦ V» - 3. r ><W S'.OO « 
-27.50 -0.5006HE-01 3.ni67l-<^. -0. 1 3246C »-)2 J.O 

11 19.00 -0.33335C-05 0.64524E*00 0.<»2963E *0<» -0. 57<.06CO3 

-27.50 -0.235756-01 -0.39101E-05 -0. 132*<SE»02 0.0 

14 9.00 -0.57829E-06 -0.61072E+02 0.10070E*05 -0. 287 liC*03 
-27.50 -0.11727E-07 3.24769E-02 -0. 13234E02 . 586 06E Ol 

15 0.0 0.0 0.64529E*00 0.100'»3E*05 3.0 
-27.50 0.92615E-02 0.0 -0. 13222C*02 0.505?U*Ol 



199 



Problem 3 - Output (continued) 



STRESSES IN CULVERT HALL (PS I) FOR LUAD INCREMENT 



NPPT 



I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

1 5 



ELLIP. OR 
INNER CAGE 
STEEL 

0.47231E+05 
0.4832 0E*05 
0. 15645E*04 
-0.42373E*03 
-3. 10902E+04 
0.0 
0.0 
0.0 
0.0 
0.0 
-0.12164E+04 
-0.65296E+03 
0. 19400EO4 
0.62375EO5 
0.62147E*05 



OUTER CAGE 
STEEL 

0.0 

■0.97052E*03 

-0.16679E*04 

0.47940E*03 

0.85525E*03 

0.27082E«-05 

0.27558E*05 

0.29503E*05 

0.29974E+05 

0.30441E*05 

0.95255E »03 

0.56926E+03 

-0.1 7701 E*04 

-0.80526E«-03 

3.0 



CONCRETE 
COMPRESSION 

■3. 247 11C + 04 
-3.2 31 01 E* 34 
-0.41377E*03 
-0. H470f>03 
-0.21280E+03 
-0.20249E*04 
-0.20582EO'. 
-0.21715E+0* 
-0.22046E*n<, 
-0.22376EO4 
■0.23742E+03 
-0. 14554E+03 
■3.43570E*O3 
-0.26233E*04 
-0.26309E*04 



SMEAR 
STRESS 

0.0 

0.42460E+02 

0.84 )73C*02 

0.86045E*02 

0.209C8E*02 

0.2306 9E*0l 

0.23019L : *01 

0.2299'»E+01 

3.22<'62E*01 

3.22999E«-0t 

-0.21608E*02 

-0. 89944E*02 

■0.90976E*02 

-0.45513E*02 

0.0 



STRAINS IN THE INNER AND OUTER FIBER Of THE CULVERT WALL 
(ONLY STRAINS FOR COMPRESSION ZONES HAVE PHYSICAL MtANING) 



NPPT 



INNER STRAIN 



OUTER STRAIN 



1 


0.1 <580E-02 


-0.47430E- 


-03 


2 


0. 19930E-02 


-3.44340E 


-03 


3 


0.80352E-04 


-0.79416E- 


-04 


4 


-0.22031E-04 


0.22609E 


-04 


5 


-0.40844E-04 


0.334 73E- 


-C4 


6 


-0.38864E-03 


0.11191E- 


-02 


7 


-0.39504E-03 


0.1138 6E- 


-02 


8 


-0.41679E-03 


0. 12176L- 


-02 


9 


-0.42314E-03 


0.12370E 


-02 


10 


-0.42947E-03 


3.12563E- 


-02 


11 


-0.45569E-04 


0.3 72B8L 


-04 


12 


-0.27935E-04 


0.27114E- 


-04 


13 


0.83477E-34 


-0.83626E- 


-04 


14 


0.23422E-02 


-0.50350E- 


-03 


15 


0.23341E-02 


-0.50496E 


-03 



200 



Problem 3 - Output (continued) 



CALCULATED SAFETY FACTORS FOR LOAD INCREMENT 5 

STEEL YIELD STRESS / MAX. STEEL STRCSS 1.59'* 

CONCRETE STRENGTH / MAX. COMPRESSIVE STRESS .... 2.647 

HALL SHEAR CAPACITY / MAX. SHEAR 1 . 835 

PERFORMANCE FACTORS 

0.01 INCH / MAX. CRACK WIDTH 0.897 



201 



APPENDIX D 



CANDE PROGRAM OVERLAY 



This appendix provides job control language (JCL) for IBM computers 
in order to reduce core storage requirements (region size) for executing 
CANDE-1980. Instructions and examples are given for two FORTRAN IV com- 
pilers commonly supported at. IBM installations; the G and the H-extended 
(HX) compilers. 

For reference, a tree chart of the CANDE-1980 subroutines is shown 
in Table D.l indicating the calling sequence of all subroutines. Sub- 
routines CANBOX and DUNCAN are new subroutines added to the CANDE-1980 
program, and subroutines CONMAT, CONCRE, and READM have been extensively 
modified from the CANDE-1976 program. Some minor changes have been made 
to other subroutines. 

Table D.2 gives the JCL to compile an overlayed version of CANDE on 
the G compiler presuming the source program resides on a disk file created 
by standard TSO operations. Here, the overlay commands (ENTRY MAIN through 
INSERT BURNS) provide a simple overlay structure that may be used as a 
guide for overlaying CANDE on most computers. 

In a similar manner, TABLE D.3 provides JCL for compiling an over- 
layed version of CANDE with the HX compiler. Here, the overlay structure 
is slightly different and takes advantage of special overlay options 
(i.e. OVERLAY C (REGION)) available at IBM installations. The overlay 
commands in Tables D.2 and D.3 may be interchanged, however the overlay 
structure in Table D.3 is more efficient on IBM. 

Once a load module is created from either Table D.2 or D.3, the 
JCL to execute the program is shown in Table D.4. Efficiency comparisons 
of executing a typical problem (Example 1, Appendix C) are shown in 
Table D.4 for the G and HX compilers with and without overlay. These 



202 



examples were executed on the IBM 370/168 computer at the University of 
Notre Dame. It is observed that HX compiler with overlay provided the 
most efficient results in terms of core storage, execution time, and 
total cost. 



203 



TABLE D.l - Subroutine Tree Structure for CANDE 



START 



CALLS 



CALLS 



CALLS 



CALLS 



CALLS 



MAIN 



ALUMIN 



BASIC 
BURNS 
CONCRE 



EMOD 
HINGE 

**PRHERO 
**PRHERO 

CONMAT 



INVER 
SETU 





**PRHERO 








PLASTI 


**PRHERO 








**PRHERO 










STEEL 


EMOD 
HINGE 

**PRHERO 


INVER 
SETU 






**PRHERO IS: 










PRHERO 


ESTAB 
HEROIC 

PREP 
RESOUT 


BAKSUB 

CONVT 

DUNCAN 

HARDIN 

INTPl 

READM 

REDUCI 
RESPIP 
STIFNS 

STRESS 

CANBOX 
CANI 

GENEL 

GENEND 

GENNOD 

MODMSH 

RESOUT 

SAVED 

PRINC 


CONVT 
CONVT 

ANISP 
CONVT 

BEAMEL 
STFSUB 
XFACES 
BEAMST 
GEOM 

XCAN2 
XCAN 

AF 
SAVEG 

PRINC 


GEOM 



J 



204 



TABLE D.2 - JCL to Create an Overlayed Version 
of CANDE Using the G Compiler 



//CNDEOVLY JOB (XX,XXXX, , 15) , IDNUMBER,NOTIFY=TSOID#, 
// REGION=256K,TIME=l 

//STEP1 EXEC FORTGCL,PARM.LKED='OVLY,MAP,XREF,LIST' 
//FORT.SYSLIN DD UNIT=DI SK,DISP=(NEW,P ASS ) ,SPACE=(TRK, (10,5) ) , 
// DSN=&&LOADSET,DCB=BLKSIZE=80 

//FORT.SYSIN DD UNIT=DISK,VOL=SER=XXXXXX,DSN=TSOID#. CANDE. FORT, 
// DISP=SHR 

//LKED.SYSLMOD DD SPACE= (1024, (600,50,1) ,RLSE) ,DISP= (NEW, CATLG) , 
// UNIT=DISK,VOL=SER=XXXXXX,DSN=TSOID#. CANDE. LOAD(CANDE) 
//LKED.SYSLIN DD UNIT=DISK,DSN=&&LOADSET,DISP= (OLD, DELETE) 
//LKED.SYSIN DD * 
ENTRY MAIN 

INSERT MAIN, PRHERO,RESOUT,PRINC,ESTAB 
INSERT STEEL, ALUMIN,EMOD 
INSERT HINGE, SETU,INVER 
INSERT C0NCRE,C0NMAT 
INSERT PLASTI, BASIC 
OVERLAY A 
INSERT PREP 
OVERLAY B 

INSERT CANB0X,CAN1,XCAN,XCAN2 
OVERLAY B 

INSERT GENNOD , SAVEG , GENEL , GENEND , AF , MODMSH , SAVED 
OVERLAY A 

INSERT HEROIC ,CONVT , GEOM, RESPIP , REDUCI , BAKSUB ,XFACES 
OVERLAY B 

INSERT READM,ANISP, DUNCAN, HARD IN, INTP1 
INSERT STIFNS , BEAMEL , STFSUB , STRESS , BEAMST 
OVERLAY A 
INSERT BURNS 
/* 



Note: Overlay commands (ENTRY MAIN through INSERT BURNS) 
start in column 2. 



205 



J 



TABLE D.3 - JCL to Create an Overlayed Version of CANDE 
Using the H Extended Compiler 



//CNDEOVLY JOB (XX,XXXX, ,15) ,IDNUMBER,NOTIFY=TSOID#,REGION=256K, 
// TIME=2 

//STEP1 EXEC F0RTXCL,PARM.F0RT='0PT(2)%PARM.LKED='0VLY,MAP,XREF,LIST' 
//FORT.SYSIN DD UNIT-DISK, VOL-SER-XXXXXX,DSN-TSOID#. CANDE. FORT, 
// DISP=SHR 

//LKED.SYSLMOD DD SP ACE= (1024, (600,50,1) ,RLSE) ,DISP= (NEW, CATLG) , 
// UNIT=DISK,VOL=SER=XXXXXX,DSN=TSOID#. CANDE. LOAD(CANDE) 
//LKED.SYSIN DD * 
ENTRY MAIN 

INSERT MAIN,PRHERO,RESOUT,PRINC,ESTAB 
OVERLAY A 

INSERT STEEL, ALUMIN,EMOD 
OVERLAY B 

INSERT HINGE, SETU,INVER 
OVERLAY A 

INSERT CONCRE,CONMAT 
OVERLAY A 

INSERT PLASTI, BASIC 
OVERLAY C (REGION) 
INSERT PREP 
OVERLAY D 

INSERT CANB0X,CAN1,XCAN,XCAN2 
OVERLAY D 

INSERT GENNOD, SAVEG , GENEL , GENEND , AF ,MODMSH , SAVED 
OVERLAY C 

INSERT HEROI C , CONVT , GEOM , RESP IP , REDUCI , BAKSUB , XFACES 
OVERLAY D 

INSERT READM,ANISP, DUNCAN, HARD IN, INTP1 
INSERT STIFNS , BEAMEL , STFSUB , STRESS , BEAMST 
OVERLAY C 
INSERT BURNS 
/* 



Note: Overlay commands start in column 2. 



206 



TABLE D.4 - JCL to run CANDE 



//CANDERUN JOB (XX,XXXX, ,10) ,IDNUMBER,NOTIFY=TSOID#, 

// REGION=256K,TIME=2 

//STEP1 EXEC PGM=CANDE 

//STEPLIB DD UNIT=DISK,VOL=SER=XXXXXX,DISP=SHR, 

// DSN-TS0ID#. CANDE. LOAD 

//FTO5F001 DD UNIT=DISK,VOL=SER=XXXXXX,DISP=SHR, 

// DSN=TSOID#.PROBNAME.DATA, 

// DCB=(RECFM=FB,LRECL=80,BLKSIZE=3120,BUFNO=1) 

//FT06F001 DD SYSOUT=A 

//FT10F001 DD UNIT=DISK,DSN=&&TEMPO,DISP= (NEW, DELETE), 

// SPACE=(TRK, (10,5)) ,DCB=(RECFM=VBS,BLKSIZE=8000,BUFNO=2) 

//FT11F001 DD UNIT=DISK,DSN=&&TEMP1,DISP= (NEW, DELETE), 

// SPACE=(TRK,(10,5)),DCB=(RECFM=¥BS,BLKSIZE=8000,BUFNO=2) 

//FT12F001 DD UNIT=DISK,DSN=&&TEMP2,DISP= (NEW, DELETE) , 

// SPACE=(TRK, (10,5)),DCB=(RECFM=VBS,BLKSIZE=8000,BUFNO=2) 

//FT13F001 DD UNIT=DISK,DSN=&&TEMP3,DISP= (NEW, DELETE) , 

// SPACE=(TRK, (10,5)),DCB=(RECFM=VBS,BLKSIZE=8000,BUFNO=2) 

//FT30F001 DD UNIT=DISK,DSN=&&TEMP5,DISP= (NEW, DELETE) , 

// SPACE=(TRK, (10,5)),DCB=(RECFM=VBS,BLKSIZE=8000,BUFNO=2) 



TABLE D.5 - Efficiency comparisons of executing CANDE 



Compiler 

& 
Overlay 


Region 

size 

(K bytes) 


Central 
Processor 

time 
(min:sec) 


Disk 
time 
(sec) 


Total 
cost 
(dollars) 












G 

No- overlay 


380 


1:43 


35.0 


$9.62 


G 
Overlay 


320 


1:44 


35.0^ 


$9.34 


HX 
No- overlay 


348 


1:03 


35.0 


$6.45 


HX 
Overlay 


264 


1:04 


35.0 


$6.14 



207 



J 



REFERENCES 

1. Katona, M.G., et al., "CANDE - A Modern Approach for the Structural 
Design and Analysis of Buried Culverts," Report No. FHWA-RD-77-5, 
Federal Highway Administration, Washington, D.C., October 1976. 

2. Katona, M.G. and Smith, J.M., "CANDE User Manual," Report No. FHWA- 
RD-77-6, Federal Highway Administration, Washington, D.C., October 
1976. 

3. Katona, M.G. and Smith, J.M., "CANDE System Manual," Report No. 
FHWA-RD-77-7, Federal Highway Administration, Washington, D.C., 
October 1976. 

4. Building Code Requirements for Reinforced Concrete, American Concrete 
Institute, ACI 318-77, 1977, Detroit. 

5. Commentary on Building Code Requirements for Reinforced Concrete, 
American Concrete Institute, ACI 318-77, 1977, Detroit. 

6. Vittes, Pedro D., "Finite Element Analysis of Reinforced Concrete 
Box Culvert," Master's Thesis, Dept. of Civil Eng. , Univ. of Notre 
Dame, May, 1980. 

7. Wang, C.K. and Salmon, D.G., Reinforced Concrete Design , Intext 
Educational Publishers, Second Edition. 

8. Sozen, M.A. and Gamble, W.L., "Strength and -Cracking Characteristics 
of Beams with #14 and #18 Bars Spliced with Mechanical Splices," 
American Concrete Institute Journal, Detroit, December 1969, pp. 
949-956. 

9. Berwanger, C. , "Effect of Axial Load on the Moment-Curvature Relation- 
ships of Reinforced Concrete Member," SP-50-11, American Concrete 
Institute, Detroit 1975, pp. 263-288. 

10. Gergely, P. and Lutz, L.A. , "Maximum Crack Width in Reinforced Con- 
crete Flexural Members," SP-20, American Concrete Institute, Detroit 
1968, pp. 87-117. 

11. Lloyd, J. P., Rejali, H.M. and Kesler, C.E., "Crack Control in One- 
Way Slabs Reinforced with Deformed Welded Wire Fabric," American 
Concrete Institute Journal, Detroit, May 1969, pp. 366-376. 

12. LaTona, R.W. and Heger, F. J. , "Computerized Design of Precast Rein- 
forced Concrete Box Culverts," Highway Research Record, Number 443, 
pp. 40-51. 

13. Boring, M.R., Heger, F.J. and Bealey, M. , "Test Programs for Evaluating 
Design Method and Standard Designs for Precast Concrete Box Culverts 
with Welded Wire Fabric Reinforcing," Transportation Research Record 
518, pp. 49-63. 



208 



3-4 • Simpson Gumpertz and Heger Inc., "Report of Test Programs for 
Evaluation of Design Method and Standard Designs for Precast 
Concrete Box Culverts with Welded Wire Fabric Reinforcing," 
submitted to American Concrete Pipe Association, July 1973. 

15. Heger, F.J. and Saba, B.K., "The Structural Behavior of Precast 
Concrete Pipe Reinforced with Welded Wire Fabric," Progress 
Report No. 2, Project No. 1-7734, Cambridge, Massachusetts, July 
1961. 

16. Heger, F.J., "A Theory for the Structural Behavior of Reinforced 
Concrete Pipe," Thesis submitted to the Department of Civil Engi- 
neering, Massachusetts Institute of Technology, January 1962. 

17. Heger, F.J., "Structural Behavior of Circular Reinforced Concrete 
Pipe-Development of Theory," Journal of the American Concrete 
Institute, November 1963, pp. 1567-1613. 

18. American Society for Testing Materials, "Standard Specification for 
Reinforced Concrete Culvert, Storm Drain, and Sewer Pipe," (ASTM 
Designation C76-70), 1970. 

19. Breton, J.M. , "Precast Box Culvert Project - Fabric Materials Test," 
Report to Frank Smith - Gifford Hill Pipe Company, April 1973. 

20. American Society for Testing Materials, "Standard Specifications 
for Welded Wire Fabric for Concrete Reinforcement," (ASTM Desig- 
nations A185-73), 1973. 

21. American Society for Testing Materials, "Standard Specification 
for Precast Reinforced Concrete Box Sections for Culverts, Storm 
Drains, and Sewers," (ASTM Designation C789-76), 1976. 

22. American Association of State Highway and Transportation Officials, 
"Interim Specification for Precast Reinforced Concrete Box Sections 
for Culverts, Storm Drains and Sewers," (AASHTO Designation: M 259- 
761), 1976. 

23. Girdler, H.F. , "Loads on Box Culverts Under High Embankments," 
Research Report 386, Department of Transportation, Division of 
Research, Lexington, Kentucky, April 1974. 

24. Russ, R.L. , "Loads on Box Culverts under High Embankments: Positive 
Projection, without Imperfect Trench," Research Report 431, Depart- 
ment of Transportation, Division of Research, Lexington, Kentucky, 
August 1975. 

25. Allen, D.L., and Russ, R.L. , "Loads on Box Culverts under High 
Embankments: Analysis and Design Considerations," Research Report 
491, Department. of Transportation, Division of Research, Lexington, 
Kentucky, January 1978. 



209 



J 



26. Kulhawy, F.H., J.M. Duncan, and H.B. Seed, "Finite Element Analysis 
of Stresses and Movements in Embankments during Construction," U.S. 
Army Eng. Waterways Experiment Station, Contract Report 569-8, 
Vicksburg, Miss., 1969. 

27. Duncan, J.M. , and C.Y. Chang, "Nonlinear Analysis of Stress and 
Strain in Soils, Journal of Soil Mechanics and Foundations Div. , 
ASCE, vol. 96, No. SM5, Sept. 1970, pp. 1629-1653. 

28. Wong, Kai S. and J.M. Duncan, "Hyperbolic Stress-Strain Parameters 
for Nonlinear Finite Element Analysis of Stresses and Movements in 
Soil Masses," Report No. TE-74-3, University of California, Berkeley, 
July 1974. 

29. Duncan, J.M. , et . al. , "Strength, Stress-Strain and Bulk Modulus 
Parameters for Finite Element Analyses of Stresses and Movements in 
Soil Masses, Report No. UCB/GT/ 78-02 to National Science Foundatidn, 
April 1978. 

30. Lee, Chee-Hai, "Evaluation of Duncan's Hyperbolic Soil Model," 
Master's Thesis, University of Notre Dame, May, 1979. 

31. Katona, M.G., £t. al . , "Structural Evaluation of New Concepts for 
Long-Span Culverts and Culvert Installations," FHWA Report No. RD- 
79-115, Washington, D.C. , December, 1979. 



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«U.S. GOVERNMENT PRINTING OFFICE: 



FEDERALLY COORDINATED PROGRAM (FCP) OF HIGHWAY 
RESEARCH AND DEVELOPMENT 



The Offices of Research and Development (R&D) of 
the Federal Highway Administration (FHWA) are 
responsible for a broad program of staff and contract 
research and development and a Federal-aid 
program, conducted by or through the State highway 
transportation agencies, that includes the Highway 
Planning and Research (HP&R) program and the 
National Cooperative Highway Research Program 
(NCHRP) managed by the Transportation Research 
Board. The FCP is a carefully selected group of proj- 
ects that uses research and development resources to 
obtain timely solutions to urgent national highway 
engineering problems.* 

The diagonal double stripe on the cover of this report 
represents a highway and is color-coded to identify 
the FCP category that the report falls under. A red 
stripe is used for category 1, dark blue for category 2, 
light blue for category 3, brown for category 4, gray 
for category 5, green for categories 6 and 7, and an 
orange stripe identifies category 0. 

FCP Category Descriptions 

1. Improved Highway Design and Operation 
for Safety 

Safety R&D addresses problems associated with 
the responsibilities of the FHWA under the 
Highway Safety Act and includes investigation of 
appropriate design standards, roadside hardware, 
signing, and physical and scientific data for the 
formulation of improved safety regulations. 

2. Reduction of Traffic Congestion, and 
Improved Operational Efficiency 

Traffic R&D is concerned with increasing the 
operational efficiency of existing highways by 
advancing technology, by improving designs for 
existing as well as new facilities, and by balancing 
the demand-capacity relationship through traffic 
management techniques such as bus and carpool 
preferential treatment, motorist information, and 
rerouting of traffic. 

3. Environmental Considerations in Highway 
Design, Location, Construction, and Opera- 
tion 

Environmental R&D is directed toward identify- 
ing and evaluating highway elements that affect 



• The complete seven-volume official statement of the FCP is available from 
the National Technical Information Service, Springfield, Va. 22161. Single 
copies of the introductory volume are available without charge from Program 
Analysis (HRD-3), Offices of Research and Development, Federal Highway 
Administration, Washington, D.C. 20590. 



the quality of the human environment. The goals 
are reduction of adverse highway and traffic 
impacts, and protection and enhancement of the 
environment. 

4. Improved Materials Utilization and 
Durability 

Materials R&D is concerned with expanding the 
knowledge and technology of materials properties, 
using available natural materials, improving struc- 
tural foundation materials, recycling highway 
materials, converting industrial wastes into useful 
highway products, developing extender or 
substitute materials for those in short supply, and 
developing more rapid and reliable testing 
procedures. The goals are lower highway con- 
struction costs and extended maintenance-free 
operation. 

5. Improved Design to Reduce Costs, Extend 
Life Expectancy, and Insure Structural 
Safety 

Structural R&D is concerned with furthering the 
latest technological advances in structural and 
hydraulic designs, fabrication processes, and 
construction techniques to provide safe, efficient 
highways at reasonable costs. 

6. Improved Technology for Highway 
Construction 

This category is concerned with the research, 
development, and implementation of highway 
construction technology to increase productivity, 
reduce energy consumption, conserve dwindling 
resources, and reduce costs while improving the 
quality and methods of construction. 

7. Improved Technology for Highway 
Maintenance 

This category addresses problems in preserving 
the Nation's highways and includes activities in 
physical maintenance, traffic services, manage- 
ment, and equipment. The goal is to maximize 
operational efficiency and safety to the traveling 
public while conserving resources. 

0. Other New Studies 

This category, not included in the seven-volume 
official statement of the FCP, is concerned with 
HP&R and NCHRP studies not specifically related 
to FCP projects. These studies involve R&D 
support of other FHWA program office research. 




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